Measurement system

Abstract

A measurement system for a quantum computing system comprising an optical coupler and a Bell state measurement apparatus. The optical coupler has N input ports and N output ports, N is greater than tw

Description

[0001] The present disclosure relates to a measurement system. The measurement system may be suitable for use m a quantum computing system comprising a plurality of quantum computing nodes. The measurement system may be suitable for performing or facilitating a Bell state measurement of photons originating from different nodes of the quantum computing system. BACKGROUND

[0002] Quantum computing utilises quantum mechanical effects such as superposition, interference and entanglement to perform computing tasks. In contrast to classical digital computers, which encode data in binary digits (bits), the basic unit of quantum information used in quantum computing is a qubit. A qubit is a unit of quantum information analogous to a bit in classical computing. However, unlike a classical bit, a qubit is subject to quantum mechanical effects and can exist in a coherent superposition of two states simultaneously.

[0003] To implement quantum computing, qubits are realised as physical qubits. A physical qubit is generally realised by maintaining and manipulating a two-state quantum mechanical physical system. For example, atoms, ions, photons, electrons and / or a superconducting electronic circuit in combination with a suitable control device may be used to realise physical qubits. A quantum computer may maintain and manipulate a plurality of physical qubits, and may encode logical qubits onto the physical qubits to execute quantum algorithms.

[0004] One of the limits to present day quantum computing is the number of physical qubits which can be maintained in a single quantum processor, which may be referred to as a core. For example, for at least some forms of physical qubits there may be a limit to the number of physical qubits which can be incorporated into a single quantum processor or core whilst maintaining the required performance and reliability of the quantum processor core.

[0005] The total number of physical qubits which can be used to execute a quantum algorithm may be increased by providing a plurality of quantum processing cores between which quantum communication links may be established. Such an implementation may be referred to as a multicore quantum computing system.

[0006] Additionally or alternatively, there may be applications of quantum computers in which a network of quantum computers is provided. Quantum communication links may be established between different quantum computers in order to enable communication of quantum states between different quantum computers in a network.

[0007] It is in this context that the subject matter contained in the present application has been devised. SUMMARY

[0008] Quantum communication links between two nodes in a quantum computing system or network (e.g., different quantum computers and / or quantum computing systems) may be established by generating photons at each of the two nodes, the photons having quantum entanglement with a qubit situated at each node. A Bell state measurement of the photons may be performed so as to achieve entanglement swapping for generating quantum entanglement between qubits situated at each of the two nodes. The generated quantum entanglement may serve as a quantum communication link between the nodes. A Bell state measurement may be performed by a measurement system arranged to receive photons originating from a plurality of different nodes. A measurement system may be arranged to receive a pair of photons originating from any of a plurality nodes in a quantum computing system and perform a Bell state measurement of the pair of photons.

[0009] The efficiency with which a Bell state measurement is performed (e.g., the proportion of attempted measurements which result in a successful Bell state measurement) may depend at least in part on optical losses in an optical path from a node at which a photon originates and a detector at which the photon is measured. For example, reducing optical losses may serve to improve the efficiency with which Bell state measurements are performed. In at least some implementations of a measurement system, optical paths between nodes in a quantum computing system and detectors at which photons originating from the nodes are measured, may include one or more switching stages and / or optical crossing points. Such switching stages and / or optical crossing points may generally increase in number with increases in a number of nodes in a quantum computing system which are served by a measurement system. Switching stages and / or optical crossing points may serve as sources of optical loss which generally degrade the efficiency with which Bell state measurements may be performed.

[0010] It has been realised that optical losses may be reduced by arranging a measurement system to include an optical circuit comprising N input ports and N output ports where N is greater than two where the optical circuit comprises an optical coupler comprising N input ports and N output ports. In particular it has been found that the use of an N x N coupler (where N is greater than two) allows for reduced optical switching and optical crossing points which can significantly reduce optical losses associated with performing a Bell state measurement of photons originating from two of a plurality of nodes of a quantum computing system. The resulting reduction in optical losses provides an improvement in the measurement efficiency with which Bell state measurements can be performed. It has further been found that an improvement in measurement efficiency is still provided even when accounting for any increase in number of detectors which may be needed (e.g., when compared to the use of a coupler having two input ports and two output ports).

[0011] According to a first aspect of the present disclosure there is provided, a measurement system for a quantum computing system, the measurement system comprising a first optical circuit comprising N input ports and N output ports, wherein N is greater than two, and wherein the N input ports are arranged to receive a first photon at a first of the N input ports and a second photon at a second of the N input ports, the first photon having quantum entanglement with a first qubit and the second photon having quantum entanglement with a second qubit and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports, and a first Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the N output ports, wherein the Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit.

[0012] The optical circuit may comprise the first optical coupler only. The N input ports and N output ports of the optical circuit may therefore each comprise the N input ports and N output ports of the first optical coupler. In such an example, the measurement system comprises a measurement system comprising a first optical coupler comprising N input ports and N output ports, wherein N is greater than two, and wherein the N input ports are arranged to receive a first photon at a first of the N input ports and a second photon at a second of the N input ports, the first photon having quantum entanglement with a first qubit and the second photon having quantum entanglement with a second qubit, and a first Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the N output ports, wherein the Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit.

[0013] In other examples, the optical circuit may comprise the first optical coupler and one or more other optical components (such as one or more further optical couplers).

[0014] The optical circuit is a passive optical circuit. That is, the optical circuit does not include any switching components which can be actively controlled to switch the path of photons.

[0015] In some examples, the input ports other than the first input port and the second input port may remain unused for a given instance of performing a Bell state measurement of a photon pair. However, the other input ports may be used for subsequent Bell state measurements of further photon pairs. For example, each of the input ports may be arranged to receive a photon of a photon pair so as to perform a Bell state measurement of the photon pair. That is, the N input ports may be arranged to receive at least the first photon and the second photon at any of the N input ports. In each instance of performing a Bell state measurement, photons may only arrive at a subset of all of the N input ports. As is described m detail herein, by providing N input ports at which photons from a photon pair may arrive, the measurement system may be arranged to perform Bell state measurements of photons originating from a plurality of different nodes with reduced optical losses.

[0016] Projecting measurements made by the plurality of single photon detectors may comprise determining a Bell state which matches the measurements. This may equivalently be thought of as mapping the measurements to a Bell state. The projection of measurements onto a Bell state may be performed with reference to a pre-determined mapping between measurement parameters (e.g., identity of detector at which photons are detected and / or times at which photons are detected) and one or more Bell states. References herein to projecting measurements onto a Bell state should be interpreted to encompass projection of measurements onto only some of all of the possible Bell states. For example, measurements of the type described herein may be used to identify (and thus perform measurements of) two of the four possible Bell states. The four possible Bell states may comprise maximally entangled states.

[0017] The first optical coupler may comprise a multimode interference coupler. Such a coupler may alternatively be referred to as a multimode interferometer. The first optical coupler comprises a passive coupler. That is, the first optical coupler does not include any switching components which can be actively controlled to switch the path of photons. An optical coupler as described herein may function as a beam splitter and may alternatively be referred to as a beam splitter.

[0018] A multimode interference coupler may comprise an optical waveguide in which light at an input port is distributed to one or more output ports. MMI couplers may be based on the selfimaging principle, by which an input field profile is reproduced in single or multiple images at periodic intervals along the propagation direction of the MMI coupler

[0019] N may be equal to four such that the first optical coupler comprises four input ports and four output ports. Similarly the first optical circuit may comprise four input ports and four output ports.

[0020] The Bell state measurement apparatus may further comprise a measurement analyser arranged to receive data from the single photon detectors indicative of measurements of photons made by the single photon detectors. The measurement analyser may be configured to determine a Bell state of the first photon and the second photon based on the data received from the single photon detectors.

[0021] The data received from the single photon detectors may be in any suitable form and may, for example, comprise digital data and / or analogue signals. For example, the data may comprise signal pulses generated by a single photon detector at a time at which the single photon detector has detected a photon. The measurement analyser may comprise any suitable electronic device, such as a classical computing device. The measurement analyser may, for example, store a preconfigured mapping of one or more measurement parameters to one or more Bell states. The measurement analyser may be configured to project measurements made by the single photon detectors (as received in the form of the data indicative of measurements of photons made by the single photon detectors) onto a Bell state using the pre-configured mapping.

[0022] The plurality of single photon detectors may be configured to measure photons in a plurality of time bins. The Bell state measurement apparatus may be configured to project the measurements made by the plurality of single photon detectors onto the Bell state based on the time bin and the single photon detector at which each of the first photon and the second photon are detected.

[0023] A single photon detector may, for example, be configured to produce a measurement indicative of a time at which a photon is detected by the single photon detector. For example, the single photon detector may output a signal at a time corresponding to a time at which a photon is detected. Such a measurement may be used (e.g., by a measurement analyser) to determine a time bin (e.g., an early time bin or a late time bin) in which the photon is detected. Which of the single photon detectors detect each photon and a time bin in which each of the photons is detected may be used to project the measurements onto a Bell state.

[0024] The Bell state measurement apparatus may be configured to project the measurements made by the plurality of single photon detectors onto the Bell state further based on the input port at which each of the first photon and the second photon are received.

[0025] A Bell state onto which a measurement is projected may depend on the input ports at which the photons arrive at the optical circuit (which may correspond with the input ports of the coupler where the circuit comprises the coupler only). The Bell state measurement apparatus (e.g., a measurement analyser which may be included in the Bell state measurement apparatus) may receive an indication of the input ports at which photons arrive at the optical circuit. The Bell state measurement apparatus may use such an indication to project measurements made by the single photon detectors onto a Bell state.

[0026] The plurality of single photon detectors may comprise N single photon detectors arranged to receive and measure photons output from each of the N output ports.

[0027] For example, a different single photon detector may be arranged to receive and detect photons output from each output port. The single photon detector at which a photon is detected may be indicative of the output port of the optical circuit from which the photon was output.

[0028] N may be greater than three (e.g., four) and the N input ports may be arranged to receive a third photon at a third of the N input ports and a fourth photon at a fourth of the N input ports, the third photon having quantum entanglement with a third qubit and the fourth photon having quantum entanglement with a fourth qubit. The optical circuit may comprise the first optical coupler and a plurality of further optical couplers arranged to provide a plurality of optical paths from each of the N input ports of the optical circuit to at least one of the N output ports of the optical circuit, wherein each of the optical paths passes through the first optical coupler. The first Bell state measurement apparatus may comprise N single photon detectors arranged to receive and measure photons output from the N output ports of the optical circuit. The Bell state measurement apparatus may be configured to project measurements made by the N single photon detectors onto Bell states so as to perform the first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit and a second Bell state measurement of the third photon and the fourth photon for generating quantum entanglement between the third qubit and the fourth qubit.

[0029] The first Bell state measurement and the second Bell state measurement may be performed in parallel with each other. For example, each of the first, second, third and fourth photons may arrive at the input ports of the optical circuit at substantially the same time. That is, the first, second, third and fourth photons may arrive at the input ports in a superposition of the same early and late time bins. In this way, the measurement system can perform two Bell state measurements of two photon pairs in parallel with each other.

[0030] The plurality of further optical couplers may comprise a plurality of 2 x 2 optical couplers each comprising two input ports and two output ports.

[0031] The optical circuit may comprise at least one phase shifter arranged in at least one of the optical paths from the N input ports of the optical circuit to at least one of the N output ports of the optical circuit.

[0032] The optical circuit may be arranged such that when each of the first, second, third and fourth photons are in a superposition of an early time bin and a late time bin, measurements made by the single photon detectors can be projected onto Bell states based on the time bins in which photons are detected and the single photon detectors at which the photons are detected.

[0033] The first Bell state measurement and the second Bell state measurement may be independent of each other and may be performed in parallel.

[0034] The measurement system may further comprise a second optical circuit comprising M input ports and M output ports, wherein M is greater than two, and wherein the M input ports are arranged to receive a fifth photon at a first of the M input ports and a sixth photon at a second of the M input ports, the fifth photon having quantum entanglement with a fifth qubit and the sixth photon having quantum entanglement with a sixth qubit, wherein the second optical circuit comprises a second optical coupler comprising M input ports and M output ports, a second Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the M output ports, wherein the second Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a third Bell state measurement of the fifth photon and the sixth photon, for generating quantum entanglement between the fifth qubit and the sixth qubit, and an optical switching network comprising a plurality of inputs each arranged to receive a photon, wherein the optical switching network is operable to selectively direct a photon received at any of the plurality of inputs to an input port of the first optical circuit or an input port of the second optical circuit.

[0035] The second optical circuit may comprise the second optical coupler only. The M input ports and M output ports of the second optical circuit may therefore each comprise the M input ports and M output ports of the second optical coupler. In other examples, the second optical circuit may comprise the second optical coupler and one or more other optical components (such as one or more further optical couplers).

[0036] The second optical circuit is a passive optical circuit. That is, the second optical circuit does not include any switching components which can be actively controlled to switch the path of photons.

[0037] M may be greater than three (e.g., four) and the M input ports may be arranged to receive a seventh photon at a seventh of the M input ports and an eighth photon at an eighth of the M input ports, the seventh photon having quantum entanglement with a seventh qubit and the eighth photon having quantum entanglement with an eighth qubit. The optical circuit may comprise the second optical coupler and a plurality of further optical couplers arranged to provide a plurality of optical paths from each of the M input ports of the second optical circuit to at least one of the M output ports of the second optical circuit, wherein each of the optical paths passes through the second optical coupler. The second Bell state measurement apparatus may comprise M single photon detectors arranged to receive and measure photons output from the M output ports of the second optical circuit. The second Bell state measurement apparatus may be configured to project measurements made by the M single photon detectors onto Bell states so as to perform the third Bell state measurement of the fifth photon and the sixth photon for generating quantum entanglement between the fifth qubit and the sixth qubit and a fourth Bell state measurement of the seventh photon and the eighth photon for generating quantum entanglement between the seventh qubit and the eighth qubit.

[0038] The third Bell state measurement and the fourth Bell state measurement may be performed in parallel with each other. For example, each of the fifth, sixth, seventh and eighth photons may arrive at the input ports of the optical circuit at substantially the same time. That is, the fifth, sixth, seventh and eighth photons may arrive at the input ports in a superposition of the same early and late time bins. In this way, the measurement system can perform four Bell state measurements of four photon pairs in parallel with each other.

[0039] The plurality of further optical couplers may comprise a plurality of 2 x 2 optical couplers each comprising two input ports and two output ports.

[0040] The second optical circuit may comprise at least one phase shifter arranged in at least one of the optical paths from the M input ports of the second optical circuit to at least one of the M output ports of the second optical circuit.

[0041] The second optical circuit may be arranged such that when each of the fifth, sixth, seventh and eighth photons are in a superposition of an early time bin and a late time bin, measurements made by the single photon detectors can be projected onto Bell states based on the time bins in which photons are detected and the single photon detectors at which the photons are detected.

[0042] The third Bell state measurement and the fourth Bell state measurement may be independent of each other and may be performed in parallel.

[0043] An optical circuit comprising more than two input ports and more than two output ports (where the number of input ports and output ports are the same) and a Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the output ports (where the optical circuit comprises an optical coupler comprising more than two input ports and more than two output ports), and configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a Bell state measurement may be referred to as a measurement unit. A measurement system may comprise a plurality of measurement units, where each measurement unit is operable to receive a pair of photons (at different input ports of the coupler) and perform a Bell state measurement of the pair of photons. The plurality of measurement units may be operable to perform Bell state measurements of different pairs of photons in parallel with each other (e.g., at the same time or at least during overlapping time periods). The use of a plurality of measurement units may therefore allow different Bell state measurements to be performed in parallel and correspondingly different quantum communication links to be established in parallel. A plurality of measurement units may therefore be used to increase a rate at which Bell state measurements are performed and correspondingly a rate at which quantum communication links can be established.

[0044] The optical switching network may be operated so as to direct photons to a desired measurement unit so as to facilitate desired Bell state measurements. For example, the switching network may be operated to direct each of a first pair of photons (between which a Bell state measurement is desired) to a first measurement unit. The switching network may be further operated to direct each of a second pair of photons (between which a Bell state measurement is desired) to a second measurement unit.

[0045] In at least some examples, the optical switching network may be fully connected. References herein to a switching network which is fully connected should be interpreted to mean that the switching network may be operated such that each of a plurality of inputs is connected (switched) to any of a plurality of outputs.

[0046] In at least some examples, the optical switching network may be be non-blocking. References herein to a switching network which is non-blocking should be interpreted to mean that in the presence of an established connection between an input of the switching network and an output of the switching network, each unused input may be directed to any of the unused outputs whilst maintaining the established connection.

[0047] The optical switching network may be arranged such that at least a first subset of a plurality of inputs to the switching network are arranged on a first side of the measurement units and a second subset of the plurality of inputs to the switching network are arranged on a second side of the measurement units. In at least some examples, a first subset of a plurality of switches of the optical switching network are arranged on a first side of the measurement units and a second subset of the plurality of switches of the optical switching network are arranged on a second side of the measurement units. In some examples, the switches may be arranged in a substantially symmetrical arrangement in which approximately half of the switches are arranged on a first side of the measurement units and a approximately half of the plurality of switches of the optical switching network are arranged on a second side of the measurement units. Such arrangements may serve to reduce the number of optical crossing points in the optical switching network.

[0048] The optical switching network may comprise a switching controller configured to control the optical switching network so as to direct a photon received at any of the plurality of inputs to an input port of the first optical circuit or an input port of the second optical circuit.

[0049] The switching controller may control the optical switching network so as to direct photons in a pair of photons for which a Bell state measurement is to be performed to the same optical circuit for performing the Bell state measurement.

[0050] The switching controller may be configured to control the optical switching network to direct the first photon received at a first of the plurality of inputs to the first of the N input ports of the first optical circuit, direct the second photon received at a second of the plurality of inputs to the second of the N input ports of the first optical circuit, direct the third photon received at a third of the plurality of inputs to the first of the M input ports of the second optical circuit, and direct the fourth photon received at a fourth of the plurality of inputs to the second of the M input ports of the second optical circuit.

[0051] The optical switching network may comprise a plurality of optical switches. At least some of the plurality of optical switches have X inputs and X outputs, wherein X is greater than two. In at least some examples, X may be equal to four such that at least some of the plurality of optical switches have four inputs and four outputs.

[0052] The at least some of the plurality of optical switches may comprise a multimode interference switch.

[0053] A multimode interference switch may comprise an optical switch which includes at least one multimode interference coupler. In at least some examples, a multimode interference switch may comprise a multimode interference coupler and a tuning apparatus for changing the selfimaging of the multimode interference coupler to achieve optical switching. The tuning apparatus may comprise at least one component (e.g., an electrode) operable to generate an electric field for inducing a change in the optical properties of the multimode interference coupler through the electro-optic effect. Changes m the optical properties of the multimode interference coupler induced through the electro-optic effect may be suitably controlled to achieve optical switching. The tuning apparatus may comprise at least one heater arranged to heat at least a portion of the multimode interference coupler. A heater arranged to heat at least a portion of an multimode interference coupler may utilise the thermoptic effect to change the self-imaging of an multimode interference coupler and achieve optical switching. The tuning apparatus may be configured to tune the multimode interference coupler using current-injection tuning to achieve optical switching.

[0054] The multimode interference switch may comprise a plurality of multimode interference couplers and at least one phase shifter situated in an optical path between the plurality of multimode interference couplers.

[0055] The at least one phase shifter may be operable to induce a phase shift in an optical path between a first multimode interference coupler and a second multimode interference coupler. In some examples, each output of a first multimode interference coupler may be optically coupled to an input of a second multimode interference coupler. A phase shifter may be arranged in each optical path between an output of the first multimode interference coupler and an input of the second multimode interference coupler. The phase shifters may be operated to achieve optical switching.

[0056] The at least some of the plurality of optical switches may comprise a plurality of optical switches having two inputs and two outputs, wherein the plurality of optical switches are arranged to function as an optical switch having X inputs and X outputs.

[0057] The plurality of optical switches having two inputs and two outputs may be arranged to provide fully connected and non-blocking switching between each of the X inputs and each of the X outputs.

[0058] References herein to fully connected switching should be interpreted to mean that the switches may be operated such that each of a plurality of inputs is connected (switched) to any of a plurality of outputs.

[0059] References herein to non-blockmg switching should be interpreted to mean that in the presence of an established connection between an input of the switches and an output of the switches, each unused input may be directed to any of the unused outputs whilst maintaining the established connection.

[0060] The plurality of optical switches having two inputs and two outputs may be arranged in the form of a Benes network of switches.

[0061] X may be equal to four such that the at least some of the optical switches have four inputs and four outputs.

[0062] According to a second aspect of the present disclosure there is provided, a measurement system for a quantum computing system comprising a plurality of measurement units comprising N input ports, and an optical switching network comprising a plurality of inputs each arranged to receive a photon, wherein the optical switching network is operable to selectively direct a photon received at any of the plurality of inputs to an input port of any of the plurality of measurement units. Each of the plurality of measurement units comprises an optical circuit comprising the N input ports and N output ports, wherein N is greater than two and wherein the N input ports are arranged to receive a first photon at a first of the N input ports and a second photon at a second of the N input ports and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports, and a Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the N output ports. The Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a Bell state measurement of the first photon and the second photon.

[0063] According to a third aspect of the present disclosure there is provided a quantum computing system comprising a plurality of quantum computing nodes, each quantum computing node being configured to maintain at least one qubit situated at the quantum computing node and wherein each quantum computing node is operable to generate a photon having quantum entanglement with the at least one qubit, and a measurement system according to any measurement system disclosed herein (e.g., a measurement system according to the first aspect or second aspect of the present disclosure), the measurement system being arranged to receive at least a first photon having quantum entanglement with a first qubit at a first of the plurality of quantum computing nodes and a second photon having quantum entanglement with a second qubit at a second of the plurality of quantum computing nodes, wherein the measurement system is operable to perform a first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit.

[0064] According to a fourth aspect of the present disclosure there is provided a method of operating a measurement system for a quantum computing system, the method comprising receiving a first photon at a first input port of an optical circuit, the optical circuit having N input ports and N output ports, wherein N is greater than two and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports, receiving a second photon at a second input port of the optical coupler, measuring, with at least one of a plurality of single photon detectors arranged at a plurality of the N output ports of the optical coupler, the first photon and the second photon, projecting the measurements of the first photon and the second photon onto a Bell state so as to perform a Bell state measurement of the first photon and the second photon.

[0065] Within the scope of this application it is expressly intended that the various aspects, embodiments, examples and alternatives set out in the preceding paragraphs, in the claims and / or in the following description and drawings, and in particular the individual features thereof, may be taken independently or in any combination. That is, all examples and / or features of any example can be combined in any way and / or combination, unless such features are incompatible. The applicant reserves the right to change any originally filed claim or file any new claim accordingly, including the right to amend any originally filed claim to depend from and / or incorporate any feature of any other claim although not originally claimed in that manner. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] One or more embodiments of the invention are shown schematically, by way of example, only in the accompanying drawings in which:

[0067] FIG. 1 is a schematic illustration of a quantum computing system including a measurement system;

[0068] FIG. 2 is a schematic illustration of an example of a measurement system for use in a quantum computing system;

[0069] FIG. 3 is a table indicating measurements of photons which correspond to a first Bell state;

[0070] FIG. 4 is a table indicating measurements of photons which correspond to a second Bell state;

[0071] FIG. 5 is a schematic illustration of a further example of a measurement system for use in a quantum computing system;

[0072] FIG. 6 is a schematic illustration of a still further example of a measurement system for use in a quantum computing system;

[0073] FIG. 7 is a schematic illustration of a still further example of a measurement system for use in a quantum computing system;

[0074] FIG. 8 is a schematic illustration of an arrangement of optical switches for use in a measurement system for a quantum computing system;

[0075] FIG. 9 is a graph illustrating optical loss in example measurement systems for a quantum computing system; and

[0076] FIG. 10 is a still further graph illustrating optical loss in example measurement systems for a quantum computing system.

[0077] FIG. 11 is a schematic illustration of a further example of a measurement unit;

[0078] FIG. 12 is a schematic illustration of a still further example of a measurement unit;

[0079] FIG. 13 is a schematic illustration of a still further example of a measurement unit;

[0080] FIG. 14 is a table indicating measurements of photons which correspond to a Bell state;

[0081] FIG. 15 is a schematic illustration of an example of measurement module;

[0082] FIG. 16 is a schematic illustration of an example of a measurement system including a measurement module; and

[0083] FIG. 17 is a further graph illustrating optical loss in example measurement systems for a quantum computing system. DETAILED DESCRIPTION

[0084] Before particular examples of the present disclosure are described, it is to be understood that the present disclosure is not limited to the particular examples described herein. It is also to be understood that the terminology used herein is used for describing particular examples only and is not intended to limit the scope of the claims.

[0085] In describing and claiming the apparatus and methods of the present disclosure, the following terminology will be used: the singular forms "a", "an", and "the" include plural forms unless the context clearly dictates otherwise. Thus, for example, reference to “a photon” includes reference to one or more of such elements.

[0086] References are made herein to a qubit and / or to a plurality of qubits. Unless indicated otherwise, such references are intended to refer to physical qubits. That is, references to qubits are intended to refer to physical entities, elements or systems which when suitably operated and controlled give rise to physical qubits. Such systems may only function as physical qubits under certain operating conditions. For example, a superconducting qubit only exhibits the behaviour of a qubit when cooled to a sufficiently low temperature such that components of the qubit exhibit superconductivity. References herein to a qubit (or physical qubit) are intended to encompass arrangements of components which are capable of functioning as qubits (for example, when cooled to suitably low temperatures) even under conditions in which they do not necessarily function as qubits.

[0087] As was explained above, in order to scale the number of qubits which can be used in quantum computations beyond the limits of a single quantum processor core, a multi-core quantum computing system may be used. A multi-core quantum computing system comprises a plurality of quantum processing cores connected via a quantum communication link. A multicore quantum computing system may be used to implement multi-qubit gates on qubits spread across a plurality of different quantum computing cores (which may be referred to as remote or distributed qubits).

[0088] Additionally or alternatively, it may be desirable to establish a quantum communication link between remotely located quantum computers or quantum computing systems. Such quantum communication links may enable a network of quantum computers or quantum computing systems, between which quantum information or quantum states can be communicated. Such a network of quantum computers may be referred to as the quantum internet.

[0089] Quantum computing cores, quantum computers and / or quantum computing systems between which quantum communication links may be established may be referred to as quantum computing nodes in a quantum computing system.

[0090] Quantum communication links may be established between nodes by creating quantum entanglement between qubits situated at different quantum computing nodes. Pairs of qubits which have quantum entanglement between them may be referred to as entangled pairs, Bell pairs or EPR pairs. Qubits which have quantum entanglement between them have quantum states which cannot be described independently of one another even when the qubits are separated by large distances. In particular, the quantum states of entangled pairs of qubits are correlated with each other.

[0091] Quantum entanglement may be created between qubits situated at different quantum computing nodes. Quantum entanglement between qubits situated at different quantum computing nodes may be established by using photons having quantum entanglement with the qubits situated at the different nodes. For example, a first photon having quantum entanglement with a first qubit situated at a first quantum computing node and a second photon having quantum entanglement with a second qubit situated at a second quantum computing node may be generated. The first and second photons may propagate over photonic communication links in order to communicate the quantum entanglement which they hold with the first and second qubits respectively. In at least some examples, the firstand second photons may arrive at a measurement station at which a Bell state measurement of the first and second photons may be performed. If the Bell state measurement is successful then entanglement swapping may be achieved, through which quantum entanglement is generated between the first and second qubits. The quantum entanglement between the first and second qubits situated at the first and second nodes respectively may form a quantum communication link between the first and second nodes.

[0092] The apparatus and methods disclosed herein relate to measurement stations for performing Bell state measurements of photons originating from different quantum computing nodes in order to establish quantum communication links between the different nodes.

[0093] FIG. 1 is a schematic illustration of a quantum computing system 102 comprising a plurality of quantum computing nodes 104 and a measurement system 106. The quantum computing nodes 104 may comprise different quantum computing cores in a multi-core quantum computing system 102. Additionally or alternatively, the quantum computing nodes 104 may comprise different quantum computers in a quantum network, which may be considered to be an example of a quantum computing system 102. Each quantum computing node 104 may establish and maintain one or more physical qubits (not shown in Fig. 1). The physical qubits may take any suitable form such as trapped ions, trapped atoms, superconducting qubits and / or photonic qubits.

[0094] A quantum communication link may be established between quantum computing nodes 104 by generating photons having quantum entanglement with physical qubits situated at different quantum computing nodes 104. The mechanism by which a photon (having quantum entanglement with a qubit situated at the quantum computing node 104) is generated at a quantum computing node 104, may depend on the type of qubit. In at least some examples, a photon may be emitted from a qubit by stimulated or spontaneous emission from the qubit. A photon emitted from a qubit may have quantum entanglement with the qubit.

[0095] In at least some examples, a qubit at a quantum computing node 104 may be situated in an optical cavity. An optical cavity may be used to capture a photon, emitted from a qubit situated inside the optical cavity, and couple the photon into a photonic link 108. For example, when a qubit is realised as a trapped ion or atom, the trapped ion or atom may be trapped in an optical cavity arranged to capture a photon emitted from the trapped ion or atom.

[0096] An optical cavity may be tuned to be at a resonant frequency which is aligned with an emission wavelength of a qubit situated inside the optical cavity. In this way a photon generation efficiency of a qubit situated inside the optical cavity may be increased by the Purcell effect. In some examples, an actuator (such as a piezoelectric transducer) may be used to adjust a length of the optical cavity so as to tune the length of the cavity. The length of the optical cavity may, for example, be tuned in response to measurements of a locking laser.

[0097] In at least some examples, a qubit may be placed in an excited energy state in order to promote emission of a photon having quantum entanglement with the qubit. For example, an excitation energy source (e.g., an excitation laser arranged to irradiate the qubit) may be used to provide energy to the qubit so as to promote excitation of the qubit to an excited energy state. A qubit in an excited energy state may subsequently relax to a lower energy state through emission of a photon from the qubit. A photon which is emitted from a qubit may have quantum entanglement with the qubit. The use of an excitation energy source to controllably place a qubit in an excited state may provide controllability over the timing at which a photon is emitted by a qubit. Additionally or alternatively, other controllable mechanisms may be used to promote emission of a photon from a qubit so as to provide controllability over the timing at which a photon is emitted by a qubit.

[0098] A photon generated at a quantum computing node 104 and having quantum entanglement with a qubit situated at the quantum computing node 104 is itself a qubit in the form of a photonic qubit. Such a photonic qubit may be transported to the measurement system 106 in order to establish quantum communication links between quantum computing nodes 104.

[0099] The quantum computing nodes 104 are photonically coupled with the measurement system 106 via photonic links 108. The photonic links 108 may comprise waveguides configured to support photons propagating between the quantum computing nodes 104 and the measurement system 106. For example, the photonic links 108 may comprise one or more optical fibres. A photon generated at a quantum computing node 104 (e.g., emitted from a qubit situated at a quantum computing node 104) may be coupled into a photonic link 108 (e.g., using an optical cavity) for propagation to the measurement system 106.

[0100] The measurement system 106 is configured to perform a Bell state measurement of at least two photons (which comprise photonic qubits) arriving at the measurement system 106. The times at which the at least two photons are generated at different quantum computing nodes 104 may be controlled such that they arrive at the measurement system 106 at substantially the same time in order to allow a Bell state measurement of the photons to be carried out at the measurement system 106. For example, qubits situated at different quantum computing nodes 104 may be excited into an excited energy state at such times that the qubits relax into a relaxed energy state and generate a photon at such times that the photons arrive at the measurement system 106 at substantially the same time.

[0101] In at least some examples, photons may be generated such that they are in a superposition of two time bins (e.g., an early time bin and a late time bin). For example, a qubit (located at a quantum computing node 104) may be subjected to an entanglement scheme which generates a photon in superposition of an early time bin and a late time bin where the superposition of the photon has quantum entanglement with the quantum state of the qubit. An example of such an entanglement scheme may comprise placing a qubit in a superposition of two ground state energy levels. A first of the ground state energy levels may then be pumped to an excited energy level from which the qubit relaxes to emit a photon. This first pumping of the first energy level may be associated with an early time bin photon emission. The populations of the ground state energy levels may then be swapped and the first ground state energy level pumped to the excited state again. This second pumping of the first ground state energy level may be associated with a late time bin photon emission. In this way the population (or wave function component) of the first ground state energy level is transferred to an early time bin photon and the population (or wave function component) of the second ground state energy level is transferred to a later time bin photon. The initial superposition of the qubit in the two ground state energy levels is thus entangled with a photon which is in a superposition of the early and late time bins.

[0102] References herein to at least two photons being generated such that they arrive at the measurement system 106 at substantially the same time may be interpreted to mean that both photons arrive at the measurement system 106 in a superposition of the same time bins.

[0103] A Bell state measurement of two photonic qubits comprises a projection of the quantum states of the two photonic qubits onto one of the four Bell states, which represent maximally entangled states. The four maximally entangled Bell states of two qubits are given below in equations (1)-(4). v2 (1) '             V2 (2) , . |01> + ^10> * ' >= J---- (3) , r ]oi> — iio> > ... J------------------- \ / 2 (4)

[0104] As was explained above, if a Bell state measurement of two photonic qubits generated at different quantum computing nodes 104 (and each having quantum entanglement with a qubit situated at the respective quantum computing node 104) is successful, then entanglement swapping may be achieved such that qubits situated at different quantum computing nodes 104 are entangled. Qubits which are situated at different quantum computing nodes 104 and which have quantum entanglement, form quantum communication links between the quantum computing nodes 104.

[0105] In the example, shown in FIG. 1, the quantum computing system 102 comprises eight quantum computing nodes 104. However, in general a quantum computing system 102 (which may comprise a quantum network and / or a multi-core quantum computing system) may include any number of nodes 104. For example, a quantum computing system 102 may include more than eight quantum computing nodes 104. It may be desirable for the measurement system 106 to be operable to perform Bell state measurements of photonic qubits originating from multiple different combinations of two of the quantum computing nodes 104 so as to allow a quantum communication link to be established between different pairs of the quantum computing nodes 104. Furthermore, it may be desirable for optical losses which occur between each quantum computing node 104 and a detector at which a photon is measured in the measurement system 106 to be reduced.

[0106] It has been realised that an efficient and low loss measurement system for performing Bell state measurements of photonic qubits originating from different quantum computing nodes 104, can be realised by incorporating an optical coupler having more than two input ports and more than two output ports in a measurement system. The optical coupler may comprise a multimode interference (MMI) coupler (which may alternatively be referred to as a multimode interferometer). In particular, it has been realised that significant advantages may be realised by incorporating an MMI coupler having more than two input ports and more than two output ports in a measurement system. An optical coupler is a passive component and does not include active switching components.

[0107] FIG. 2 is a schematic illustration of a measurement system 202 according to examples of the present disclosure. The measurement system 202 may be used in a quantum computing system to perform Bell state measurements of photons originating from different nodes of the quantum computing system. For example, the measurement system 202 may be suitable for performing Bell state measurements of photons originating from different quantum computing nodes 104 of a quantum computing system 102 having any of the features of a quantum computing system 102 described above with reference to FIG. 1. The measurement system 202 described with reference to FIG. 2 may, for example, form all or part of the measurement system 106 described above with reference to FIG. 1.

[0108] The measurement system 202 comprises an optical coupler 204 and a plurality of single photon detectors 210. The coupler 204 may comprise an MMI coupler 204. An MMI coupler is an optical waveguide in which light at an input port is distributed to one or more output ports. MMI couplers are based on the self-imaging principle, by which an input field profile is reproduced in single or multiple images at periodic intervals along the propagation direction of the MMI coupler. A detailed description of the principles of an N x N MMI coupler is provided, for example, by [1], The MMI couplers contemplated herein generally comprise a plurality of input ports and a plurality of output ports. In particular, the MMI couplers contemplated herein comprise symmetrical couplers comprising N input ports and N output ports. According to examples disclosed herein N is greater than two.

[0109] In the example depicted in FIG. 2, the coupler 204 comprises a coupler comprising four input ports 206 and four output ports 208 (i.e., N=4). Each of the input ports 206 are arranged to receive a photon. For example, each of the input ports 206 may be arranged to receive a photon from a quantum computing node 104 in a quantum computing system 102. The single photon detectors 210 are each arranged to receive and measure single photons output from a respective output port 208. The lines representing the input ports 206, output port 208 and joining the output ports 208 to the single photon detectors 210 represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0110] The single photon detectors 210 may comprise any suitable form of single photon detector 210. For example, the single photon detectors 210 may comprise a superconducting detector such as a superconducting nanowire single-photon detector or a semiconductor-based single-photon detector. In at least some examples, the single photon detectors 210 may be of a type having a relatively short dead-time after detection of a first photon and before a second subsequent photon can be detected. Using a single photon detector 210 with a relatively short dead-time may allow the same single photon detector 210 to detect photons in multiple time bins (e.g., in both an early and a late time bin as will be described in more detail below).

[0111] It has previously been demonstrated [2] that quantum interference can be realised in an MMI coupler. It has been realised by the present inventors that an N x N MMI coupler (where N is greater than two) may be used to perform Bell state measurements of photonic qubits with reduced optical losses.

[0112] A coupler 204 (such as an MMI coupler 204) may be incorporated in a measurement system 202 for performing Bell state measurements of photonic qubits whose quantum states are encoded in time bins. Previous measurement approaches to performing Bell state measurements of photonic qubits have largely focussed on photonic qubits whose quantum states are encoded in the polarisation states of the qubits. In the examples contemplated herein, the quantum state of photonic qubits may be encoded in time bins. In such examples, the orthogonal states of a photonic qubit may comprise an |earZy> = :0,> sta|e anj a \late> = |1> state. A photon generated at a quantum computing node 104 and having quantum entanglement with a qubit situated at the quantum computing node 104 may be considered to be in a superposition of the \ear^y) and states. Using such an encoding, a Bell state measurement of photons arriving at the measurement system 202 may be carried out according to the identity of the single photon detector(s) 210 at which the photons are detected, the time bin(s) in which the photons are detected and the identity of the input ports 206 at which the photons are received.

[0113] As was explained above, each of the input ports 206 may be arranged to receive a photon generated by a quantum computing node 104. For example, to provide a simple illustrative example, the measurement system 202 shown in FIG. 2 could be used in a quantum computing system comprising four quantum computing nodes 104. Each of the four input ports 206 of the measurement system 106 may be arranged to receive a photon from a different one of the four quantum computing nodes 104.

[0114] In such an example, a quantum communication link between two of the four quantum computing nodes 104 may be established by generating a first photon at a first quantum computing node 104 and generating a second photon at a second quantum computing node 104. As was explained above, the first photon may have quantum entanglement with a qubit situated at the first quantum computing node 104 and the second photon may have quantum entanglement with a qubit situated at the second quantum computing node 104. The first and second photons may each comprise photonic qubits in a superposition of an \^arly> anc[ \late) state.

[0115] The first and second photons may be generated at the first and second quantum computing nodes 104 such that they arrive at the measurement system 202 at substantially the same time. For example, qubits at the first and second quantum computing nodes 104 respectively may each be excited into an excited energy state at such times that each of the qubits relax their energy states and generate photons at such times that the photons arrive at the measurement system 202 at substantially the same time.

[0116] Whilst photons are described herein as arriving at a measurement system at substantially the same time, it will be appreciated that, as explained above, the photons may be in a superposition of temporal states (time bins) such that measurement of the photons suggests that the photons may have arrived at different times. References herein to photons arriving at a measurement system at substantially the same time should be considered to encompass photons which arrive in a superposition of different temporal states. For example, references to at least two photons being generated such that they arriving at substantially the same time may be interpreted to mean that both photons arrive in a superposition of the same temporal states (time bins).

[0117] The first and second photons arrive at different input ports 206 of the coupler 204 and may be detected by at least one of the single photon detectors 210 and in at least one of an early time bin and a late time bin. The measurements made by the single photon detectors 210 may be projected onto a Bell state based on the time bm(s) in which the photons are detected, the single photon detector(s) 210 at which the photons are detected and the input ports 206 at which the photons are received. It has been found that two photons arriving at different input ports 206 of a coupler 204 can be used to discriminate between two of the four Bell states. In particular, as will be shown in more detail below, two photons arriving at different input ports 206 of an N x N coupler 204 can be used to discriminate between the |'1>+and \ Bell states.

[0118] In order to show that a measurement of photons output from an N x N coupler 204 can be used to discriminate between the [¢ + ) and j Bell states, a transformation produced by an interferometer (in this case an N x N coupler 204) may be represented by a unitary matrix #2k. The unitary matrix -¾½ may be constructed recursively as -¾ ® where = -i- [ ) and k = / -,- For example, in many of the examples V2 V -V / 2 presented herein a 4 x 4 coupler 204 is contemplated such that N = 4. In such examples, is a 4 x 4 matrix constructed as ^2 ® H2.

[0119] In the case of an N x N coupler, a photon pair can arrive at any two of N inputs (e.g., input ports 206). This may be denoted using indices i and j representing the input at which a first photon and a second photon arrive respectively, where i and j are integers from 1 to N. In general, we consider photon pairs where the photons arrive at different inputs such that $ 3. Using this notation the and I¢ ) Bell states may be written as shown in equations (5) and (6) below. (6)

[0120] Equations (5) and (6) represent all of the possible input states of a pair of photons arriving at inputs i and j respectively which correspond to one of the Bell states + \. . . Using this notation it can be shown that a coupler 204 is capable of discriminating between the Bell states . undjM> ). . (and therefore perform a measurement of those Bell states) if the transformation induced by the unitary matrix -¾ on the ij modes of A. ;andVpleads to orthogonal vectors such that they can be uniquely distinguished from each other .

[0121] In order to demonstrate this result, |^+). .and are rewritten in terms of the annihilation and creation operators such that As was explained above, i and j are indices denoting the inputs to the coupler 204 (and equivalently the input modes of the unitary matrix -¾). The indices k and 5 are used here to label the variable used to encode the photonic qubits and are the integers 1 or 0. In examples contemplated herein, photonic qubits may be encoded in time bins. Expressions of the Bell states ,and|^"^ in terms of the annihilation and creation operators are shown in equations (7) and (8) below.

[0122] Using this notation, the interferometer (coupler 204) represented by the unitary matrix -¾ “ H induces a transformation on the field operators as shown in equation (9) in which the label 2k has been removed from the Unitary H to properly reflect the dimensionality of the interferometer. jV HisO-ks s-.l (9)

[0123] The operators akb afamay be referred to as input and output field operators respectively. The evolution of the states i .and ,4' "). . can then be described by replacing the input field operators with respective output field operators. The evolution of ).. ?.and |$~) can then be constructed by evaluating the expressions and which are shown in equations (10) and (11) below. N N N igH:jSaisaas + s—1 a;—1 y—l.x^y V V HiXH3ya^ (10) (11)

[0124] In equations (10) and (11) above the terms are grouped in the multiplication so as to separate terms that come from the diagonal (same input different time bin) and those that come off diagonal (different input and different time bin). Using the properties of the unitary matrix three sets of indexes can be defined as shown in equations (12), (13) and (14) below. (12) (13) (14)

[0125] By using the indexes defined in equations (12), (13) and (14), equations (15) and (16) below follow. N T HixHjsaisaQa 4- T 0.0,^(2]^) (15) (iQid ij — — (16)

[0126] By applying the operator expressions in (15) and (16) to the vacuum state (and using 7+ n 7~ — 0 and 7." n 7° — 0) it can be seen that the resulting vectors corresponding to the evolution of [¢+) and [T ).. are orthogonal to each other such that the 1'1' and . states can be uniquely distinguished from each other through measurement of photons output from a coupler 204 represented by ^hk as described above.

[0127] Using the indexes defined above it is then possible to evaluate the sets It. , I~ and / ° and use them to identify, for each possible set of input pairs ij, the measurements which correspond to the and states. This evaluation can be carried out recursively starting from the case in which N = 2 corresponding to a 2 x 2 coupler. In this case 70 = [(i, ^2, 2), (3,3), (4,4)] an£l "*s emPly- The first case for which is non-empty is N ~ 4 corresponding to a 4 x 4 coupler 204 as shown in FIG. 2. For N ~ 4, the coupler 204 is represented by a unitary matrix H4 fl 1 1 11 (17)

[0128] By looking at the matrix given in (17) the pairs of indices which are in the set I + can be found. For example, a pair of indices can be chosen and the corresponding submatrix (corresponding to the chosen index pair) evaluated to find the pairs of indices corresponding to rows of the submatrix which are the same ( ^1,^2 for which the couples (Ht kl. HjM ), (Hi are the same). For example, if 3 — 1; 2 is chosen then the corresponding submatrix is given by (18) below. p M i -i i i .1 -I. (18)

[0129] Evaluation of the submatrix shown in (18) shows that the pairing of row indices which are the same are (1? 3) and (2? 4).

[0130] In the case of A = 4 corresponding to a 4 x 4 coupler 204, there are six unique possible pairs of different inputs ignoring pairs in which the order is swapped and excluding pairs with equal indices (since we only consider photon pairs arriving at different inputs). It has been found that three input pairs corresponding to (1» ^), (1.3), (1,4) can he evaluated initially and the other input pairs can be identified based on these input pairs by adding 2 to each index. This relationship results because the pair of columns of Hi are the same under the operation (~'2, modulo 4) on coiumn index such that columns (1,2) will iead to the same pairing as (4,3) and equivalently for other pairs (except for (l^) which is paired with (2,4)) By using this method, the elements of / .t are as given in (19) below. 7^ = [(1,3),(2,4)] 7^ = ((1,3),(2,4)] 7^ = ((1,2),(3,4)] 7- ((1,2),(3,4)] 7^ = ((1,4),(2,3)] = ((1,4),(2,3)] (19)

[0131] The elements of can then be evaluated by exclusion and knowing all possible pairs. This method provides / j as given in (20) below. - [(1,2),(3,4),(1,4),(2,3)] I,, - [(1.2),(3,4),(1,4),(2,3)] / 13-((1,3),(2,4),(1,4),(2,3)] [(1,3),(2,4), (1,4),(2,3)] = [(1,3),(2,4),(1,2),(3,4)] / 32 — [(1,3), (2,4), (1,2), (3,4)] (20)

[0132] The evaluations of / 4 and / .” as shown in (19) and (20) above can be used to identify all measurements for a photon pair at a given pair of inputs of a 4 x 4 coupler 204 which correspond to the [¢+) and ¢-) Bell states. FIG. 3 is a table representing all of the measurement combinations which correspond with the 4 ) Bell state for pairs of photons arriving at each of the six combinations of input port pairs at which two photons can arrive. The different input port pairs are shown in the left-most column of the table of FIG. 3 and the different output port pairs (at which a pair of photons might be detected) are shown in the upper-most row of the table of FIG. 3. The crosses marked in the table of FIG. 3 indicate (for photons arriving at each input port pair) the output port measurement pairs which correspond with the |^ +) Bell state. Two crosses m a single box in the table of FIG. 3 indicate measurements of two photons in different time bins (as denoted by tO and tl in the headings in the second row of the table of FIG. 3).

[0133] It can be seen from FIG. 3 that for photons arriving at any combination of the input ports, if a photon is detected in the early time bin at a first output port 208 (by a first single photon detectors 210) and a photon is detected in the late time bin at the same first output port 208 (by the same first single photon detector 210) then the measurement is projected onto the [¢ + ), Bell state. That is, detecting two photons in different time bins and at the same single photon detector 210 corresponds to the ’ \ Bell state.

[0134] It can also be seen from FIG. 3 that for each input port combination, there are measurements of photons at different output ports 208 (by different single photon detectors 210) and in different time bins which also correspond to the [¢ + ) Bell state. For example, for a pair of photons arriving at input ports 1 and 2 (input port indices a measurement of a first photon in a first time bin at output port 1 and a second photon in a second time bin (different to the first time bin) at output port 3 (output port indices (1=2) anc[ (3,1)) aiso corresponds to the ) Bell state. Similarly, a measurement of a first photon in a first time bin at output port 2 and a second photon in a second time bin (different to the first time bin) at output port 4 (output port indices (2, 4) ancj (4, 2)) aiso corresponds to the [¢+ ) Bell state.

[0135] The further measurements which correspond to the Bell state for photon pairs arriving at other pairs of input ports 206 can be evaluated by considering the positions of the crosses in FIG. 3 for the required pair of input ports 206.

[0136] FIG. 4 is a table representing all of the measurement combinations which correspond with the | $ ) Bell state for pairs of photons arriving at each of the six pairs of input ports. The format of FIG. 4 is the same as the format of FIG. 3 and no detailed description of this format will be provided again with reference to FIG. 4. It can be seen from FIG. 4 that for pairs of photons arriving at each pair of input ports 206 there are measurement combinations of photons in different time bins and at different output ports 208 which correspond with the ; ) Bell state. It can be further seen that the combinations of output ports 208 for which a measurement of photons in different time bins corresponds to the |4< ) Bell state are different to the combinations of output ports 208 for which a measurement of photons in different time bins corresponds to the Bell state. This therefore confirms that a 4 x 4 coupler 204 can be used to uniquely distinguish between the and Bell states.

[0137] As has been described above, and illustrated by FIG. 3 and FIG. 4, a measurement of two photons in different time bins can be projected on to either the |$ ! ) Bell state or the Bell state in dependence on which of the single photon detectors 210 the photons are detected at and based on knowledge of which of the input ports 206 the photons arrived at. The projection of the measurement of two photons in different time bins onto one of the ) Bell state or the ) Bell state may be carried out with reference to the mapping illustrated in FIG. 3 and FIG. 4.

[0138] As was described above, it has been shown that a 4 x 4 coupler 204 (e.g., of the type illustrated in FIG. 2) can be used to project a measurement of a pair of photons (arriving at any two different input ports 206 of the four input ports 206) output from one or more of the output ports 208 of the coupler 204 and in different time bins onto one of two Bell states. It has also been found that a similar projection can be made for N >4 using corresponding principles to those described above for the A; =--- 4 case.

[0139] For each value of N, and for each pair of input ports of a coupler, pairs of output ports (output pairs) may be identified at which a measurement of photons would correspond to one of two Bell states (in particular the [¢ + ) and (¢ ) Bell states). This identification may be carried out recursively building on identified pairs for smaller values of N . For example, for a given value of N, pairs may be identified starting from pairs which were identified for an N case. A' For each value of A’, there are ——2__ unique combinations of two different input ports (input pairs). However, in practice only half of these input pairs need to be considered because half of the input pairs correspond with the other half of the input pairs but with indices swapped, which leads to the same statistics. Furthermore, in practice only half of these pairs again need be considered since the sub-matrices * / 2] 3re Per*os^c along c°lumn or row whh modulo A / and the sub-matrices / ¾ ] A / 4. 1 : ;V are periodic (with the sign flipped) along column or row with modulo Consequently, it is possible to only identify output port pairs N\ for -sets of input pairs and find output port pairs for the other input port pairs by symmetry and exclusion.

[0140] In some examples, output pairs may be identified initially for the input pairs $), (1,3)..... (1. Ar) jf qie pajrs are identified recursively such that output pairs have previously been identified for smaller A' and in particular for then at least some of the output pairs may already be know. In particular, where output pairs have been previously identified for an A / case then previously identified pairs will be present in the top left (denoted tl herein) A / x N / block of the unitary H x. There are then relevant sets of column indices ij. For each of these sets A^ pairs of row indices may be found for which k., Hj k,) ~ (Hi Hj ■ These pairs may be expressed as belonging to a set 5^ given by (21) below, where the labels + and - are used to distinguish between pairs of even and odd parity. (21)

[0141] Pairs in a down-left (denoted dl herein) block of the unitary -¾ can then be identified by adding the indexes ku and ^2«. This set of pairs may be expressed as belonging to a set s^given by (22) below. u = + -y, &21 + 4")«» • • • + + 4-) , ( k\ / * ...p + -v, Z?2 + ¼) , • • • ( + "T; ^2 A + ) \ X h 1 / \ $ / M / i j x 4 ~ ~ / i j (21)

[0142] To generate all of the remaining output pairs, all the combinations of indices between the two sets and and between the two sets and may be taken. Combinations in which indices are swapped and combinations in which the indices are equal may be discarded. This may be expressed as finding the set given in equation (22) below in which (J± is the set of all of the combinations of indices between the sets and s(U±. if - (S“) U (Sj) u (cl) u (cf (22)

[0143] Where the pairs are identified recursively based on pairs identified for small N it is noted that 5^ — and then all other sets may be generated from

[0144] As was explained above for the N ~ 4 case, the members of / " ’,vmay be identified based on [pausing the fact that all possible pairs of indices are given by the union expressed in equation (23) below where ~ [(1, 1), (2,2),... [Ar, • u u y v J \ W / \ ^ / (23)

[0145] As has been demonstrated above, for an N x N coupler (which may, for example, comprise an MMI coupler), for each possible combination of different input ports (input pairs) at which a pair of photons may arrive, one or more first pairs of output ports (first output pairs) may be identified which correspond with a measurement of a Bell state and one or more second pairs of output ports (second output pairs) may be identified which correspond with a measurement of a (¢) Bell state. The one or more first output pairs correspond with pairs of output ports at which a measurement of a first photon in a first time bin at one of the output ports in the respective pair and a measurement of a second photon in a second time bin (different to the first time bin) in the other of the output ports in the respective pair represents a measurement of the Bell state. Similarly, the one or more second output pairs correspond with pairs of output ports at which a measurement of a first photon in a first time bin at one of the output ports in the respective pair and a measurement of a second photon in a second time bin (different to the first time bin) in the other of the output ports in the respective pair represents a measurement of the [¢-Bell state.

[0146] Using the identifications of output pairs described above, for a given N x N coupler, a measurement of a first photon in a first time bin and at a first output port and a measurement of a second photon in a second time bin (different to the first time bin) and at a second output port (which may be different to or the same as the first output port) may be mapped on to either the | Bell state or the Bell state. Such a mapping of a measurement to a Bell state (e.g., either the Bell state or the Bell state may be referred to as a projection onto a Bell state so as to perform a Bell state measurement.

[0147] Returning again to the measurement system 202 of FIG. 2 (which represents an Ar — 4 case), the single photon detectors 210 of the measurement system 202 may be considered to form part of a Bell state measurement apparatus configured to measure photons and project the measurements made by the single photon detectors 210 onto a Bell state. The Bell state measurement apparatus comprises at least the single photon detectors 210. The Bell state measurement apparatus may further comprise a measurement analyser 212. The measurement analyser 212 may be communicatively coupled to the single photon detectors 210 and configured to receive data from the single photon detectors 210 indicative of the measurements made by the single photon detectors 210. The data may be in any suitable form and may, for example, comprise digital data and / or analogue signals. For example, the data may comprise signal pulses generated by a single photon detector 210 at a time at which the single photon detector 210 has detected a photon.

[0148] The measurement analyser 212 may be configured to determine a Bell state of two photons based on the measurements made by the single photon detectors 210. In particular, the measurement analyser 212 may be configured to determine a Bell state of two photons based on which of the single photon detectors 210 detect the photons (and hence the output ports 208 at which the photons are measured), the time bins in which the photons are detected and the input ports 206 at which the photons arrive. As was explained above, for each pair of input ports 206 at which photons arrive, a first set of pairs of output ports 208 may be pre-determined at which photon measurements correspond to the | +) Bell state and a second set or pairs of output ports 208 may be pre-determined at which photon measurements correspond to the \ Bell state. The pre-determined mapping of output port 208 measurements to the [¢ + / Bell state for each combination of different input ports 206 for a 4 x 4 coupler 204 is shown in the table of FIG. 3. The pre-determined mapping of output port measurements to the ) Bell state for each combination of different input ports 206 for a 4 x 4 coupler 204 is shown in the table of FIG. 4. The measurement analyser 212 may be configured to project measurements made by the single photon detectors 210 onto a Bell state using the pre-determined mapping of output port 208 pairs for each combination of input ports 206 (as was described above), such that a Bell state measurement is performed.

[0149] The measurement analyser 212 may comprise any suitable electronic device, such as a classical computing device. The measurement analyser 212 may be communicatively coupled to one or more other components of a quantum computing system 102. For example, the measurement analyser 212 may communicate the result of a Bell state measurement to one or more of the quantum computing nodes 104 and / or a control device responsible for establishing quantum communication links between the quantum computing nodes 104.

[0150] As described above with reference to FIG. 2, a 4 x 4 coupler 204 may be used in a measurement system 202 to perform a Bell state measurement of two photons arriving at two of the four input ports 206 of the coupler 204. The two photons arriving at two of the four input ports 206 may correspond to photons generated at two different quantum computing nodes 104. A successful Bell state measurement performed by the measurement system 202 may therefore be used to establish a quantum communication link between two different quantum computing nodes 104. The use of a 4 x 4 coupler 204 as described above with reference to FIG. 2 therefore allows the establishment of a quantum communication link between any two of four quantum computing nodes 104 and without the need for any optical switching.

[0151] The use of a 4 x 4 coupler 204 as described above with reference to FIG. 2 may be compared to an equivalent arrangement in which a 2 x 2 coupler and two single photon detectors is used to perform a Bell state measurement. In such an arrangement, in order to be able to establish quantum communication links between any of four quantum computing nodes 104 then optical switching would be required in order to switch photons from two of the four quantum computing nodes 104 to arrive at two of the input ports of a 2 x 2 coupler.

[0152] The choice of, for example, a 4 x 4 coupler 204 to perform a Bell state measurement of two photons may be considered to be counterintuitive when compared to the use of a 2 x 2 coupler, since a 4 x 4 coupler 204 requires four single photon detectors 210 to achieve the same measurement efficiency as a 2 x 2 coupler. However, as will be explained in more detail below it has been found that the use of a 4 x 4 coupler 204 (and more generally an N x N coupler where N is greater than two) allows for reduced optical switching and optical crossing points which can significantly reduce optical losses associated with performing a Bell state measurement of photons originating from two of a plurality of quantum computing nodes 104 in a quantum computing system 102. The resulting reduction in optical losses provides an improvement in the measurement efficiency with which Bell state measurements can be performed.

[0153] As was explained in detail above with reference to FIG. 2, the measurement system 202 may be used to perform Bell state measurements of photons originating from any of four different quantum computing nodes 104. The principals described with reference to FIG. 2 can be scaled up to larger measurement systems capable of performing Bell state measurements of photons originating from more than four different quantum computing nodes 104. For example, as will be described in further detail below with reference to FIG. 5 - FIG. 7, the measurement system 202 of FIG. 2 may be used as a building block in larger measurement systems capable of performing Bell state measurements between photons originating from larger numbers of quantum computing nodes 104. In subsequent examples, the measurement system 202 of FIG. 2 may be referred to as a measurement unit which is included in a larger measurement system.

[0154] FIG. 5 is a schematic illustration of a measurement system 502 according to further examples of the present disclosure. The measurement system 502 comprises two measurement units 508. The measurement units 508 shown in FIG. 5 include corresponding components to the measurement system 202 described above with reference to FIG. 2. In particular, each measurement unit 508 comprises a coupler 204, and a Bell state measurement apparatus comprising a plurality of single photon detectors 210. As was described above with reference to FIG. 2, the plurality of single photon detectors 210 form part of a Bell state measurement apparatus configured to project measurements made by the single photon detectors 210 onto a Bell state. Whilst not shown in FIG. 5, one or more of the measurement units 508 may further comprise a measurement analyser 212 as was described above with reference to FIG. 2.

[0155] Corresponding components of the measurement units 508 depicted in FIG. 5 and the measurement system 202 depicted in FIG. 2 are provided with corresponding reference numerals. For ease of illustration, reference numerals may not be provided for each and every component of the measurement system 502 shown in FIG. 5. For example, only some of the single photon detectors 210 are explicitly labelled with reference numerals in FIG. 5. The measurement units 508 shown in FIG. 5 may include any of the features or components of the measurement system 202 described above with reference to FIG. 2. No further detailed description of the operation of the measurement units 508 will therefore be provided with reference to FIG. 5.

[0156] Each of the measurement units 508 are configured to perform a Bell state measurement of two photons arriving at two of four input ports 206 of the coupler 204 which is included in each measurement unit 508. The measurement units 508 may operate independently and may, for example, be operable to perform independent Bell state measurements simultaneously. The measurement units 508 may therefore be used to perform two Bell state measurements in parallel so as to generate two quantum communication links at the same time.

[0157] It may be desirable for the measurement units 508 to perform Bell state measurements between photons originating from different combinations of two quantum computing nodes 104 of a quantum computing system 102. In particular, it may be desirable to direct two photons originating from different pairs of quantum computing nodes 104 to the same measurement unit 508 such that a Bell state measurement of the two photons can be performed at the measurement unit 508.

[0158] The measurement system 502 further includes an optical switching network comprising a plurality of optical switches 504. The optical switching network generally comprises a plurality of inputs 506 at which photons can be received and is configured to selectively direct a photon received at each of the plurality of inputs 506 to one of the measurement units 508. In the example depicted in FIG. 5, the measurement system 502 comprises a first measurement unit 508 and a second measurement unit 508. The optical switching network comprises switches 504 which are operable to direct a photon received at any of the inputs 506 to either the first measurement unit 508 or the second measurement unit 508. In particular, the optical switching network is operable to direct a photon received at any of the inputs 506 to an input port of a coupler 204 included in the first measurement unit 508 or an input port of a coupler 204 included in the second measurement unit 508. In this way, the optical switching network may be operated to facilitate establishing quantum communication links between different pairs of a plurality of quantum computing nodes 104. In the example measurement system 502 shown in FIG. 5, the optical switches 504 may be operated to facilitate a Bell state measurement between a pair of photons arriving at any two of the inputs 506, except for two inputs 506 of the same switch 504. That is, the switches 504 are operable such that two photons arriving at any two of the inputs 506, where the two inputs 506 are inputs 506 of different switches 504, may be directed to the same measurement unit 508 to facilitate a Bell state measurement between the two photons. Furthermore, the switches 504 are operable to facilitate two Bell state measurements between two pairs of photons arriving at any of the inputs 506 (where each pair of photons arrives at a different switch 504) m parallel.

[0159] The lines joining components shown in FIG. 5 (e.g., lines joining switches 504 and couplers 204) represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0160] The optical switches 504 may comprise any suitable form of optical switch. In at least some examples, the optical switches may comprise optical switches based on an MMI coupler. An optical switch based on an MMI coupler may comprise an MMI coupler and at least one heater arranged to heat at least a portion of the MMI coupler. A heater arranged to heat at least a portion of an MMI coupler may utilise the thermooptic effect to change the self-imaging of an MMI coupler and achieve optical switching. Principles behind an optical switch based on an MMI coupler are described, for example, m [3], In other examples, an optical switch based on an MMI coupler may comprise an MMI coupler which is tuned using current-injection tuning to achieve optical switching. Optical switches which are based on an MMI coupler may be described herein as an MMI switch.

[0161] In the example depicted in FIG. 5 the optical switching network comprises a plurality of 2x2 optical switches 504. That is, each of the switches 504 comprises two inputs and can switch between two outputs. Each switch 504 has a first output coupled with a first of the measurement units 508 and a second output coupled with a second of the measurement units 508. Each input 506 can therefore be switched between either the first of the measurement units 508 or the second of the measurement units 508. In this way the switches 504 may be operated to direct a first pair of photons received at two of the inputs 506 to the first of the measurement units 508 and / or a second pair of photons received at two of the inputs 506 to the second of the measurement units 508. The measurement system 502 is therefore operable to perform two Bell state measurements and create two quantum communication links in parallel.

[0162] In the example of FIG. 5, the measurement system 502 comprises eight inputs 506. In some examples, each input 506 may be coupled to a different quantum computing node 104 (not shown in FIG. 5). The measurement system 502 may therefore be operable to establish quantum communication links between one or more pairs of the eight quantum computing nodes 104.

[0163] Each optical path from an input 506 to a measurement unit 508 passes through one optical switch 504. The measurement system 502 of FIG. 5 may therefore be described as including one switching stage. As can be seen in FIG. 5, some of the optical paths from a switch 504 to a measurement unit 508 includes a crossing point 510 at which two optical paths cross.

[0164] Switches 504 and crossing points 510 are both known to be sources of optical loss. Reducing the number of switches 504 and / or crossing points 510 in an optical path between an input 506 and a measurement unit 508 can therefore reduce optical loss along that path. As will be shown in further detail below, the arrangement of the measurement system 502 of FIG. 5 has been found to exhibit lower optical losses than other comparable arrangements.

[0165] In the arrangement shown in FIG. 5, half of the switches 504 are arranged on a first side (the left-hand side in FIG. 5) of the measurement units 508 and half of the switches 504 are arranged on a second side (the right-hand side of FIG. 5) of the measurement units 508. By positioning switches on both of two sides of the measurement units 508 (e.g., the symmetrical arrangement shown in FIG. 5 in which half of the switches are positioned on each side), the number of crossing points 510 may be minimised.

[0166] As was explained above, the measurement system 502 of FIG. 5 is operable to receive photons from any of eight inputs 506 and is operable to perform two Bell state measurements (to generate two quantum communication links) at two measurement units 508 in parallel. The arrangement of FIG. 5 can be further scaled up to include further switching stages (to increase the number of inputs) and / or further measurement units 508 (to increase the number of Bell state measurements which can be performed). In at least some examples, the measurement system 502 may be used as a building block in larger measurement systems including a larger number of inputs (and therefore capable of performing Bell state measurements between photons originating from larger numbers of quantum computing nodes 104). In subsequent examples, the measurement system 502 of FIG. 5 may be referred to as a 2-pair measurement unit (called as such because it can carry out two Bell state measurement between two pairs of photons) which may be included in a larger measurement system.

[0167] FIG. 6 is a schematic illustration of a measurement system 602 according to further examples of the present disclosure. The measurement system 602 comprises four measurement units 508. The measurement system 602 may also be considered to comprise two 2-pair measurement units 604. Each 2-pair measurement unit 604 includes corresponding components to the measurement system 502 described above with reference to FIG. 5. In particular, each 2-pair measurement unit 604 includes two measurement units 508 and a switching stage comprising four switches 504.

[0168] Corresponding components of the measurement units 508 depicted in FIG. 6, the measurement units 508 depicted in FIG. 5 and the measurement system 202 depicted m FIG. 2 are provided with corresponding reference numerals. Furthermore, corresponding components of the 2-pair measurement units 604 depicted in FIG. 6 and the measurement system 502 depicted in FIG. 5 are provided with corresponding reference numerals. For ease of illustration, reference numerals may not be provided for each and every component of the measurement system 602 shown in FIG. 6. The measurement units 508 of the measurement system 602 of FIG. 6 may include any of the features or components of the measurement system 202 described above with reference to FIG. 2 and / or the measurement units 508 described above with reference to FIG. 5. Furthermore, the 2-pair measurement units 604 of the measurement system 602 of FIG. 6 may include any of the features or components of the measurement system 502 described above with reference to FIG. 5. No further detailed description of the operation of the measurement units 508 or 2-pair measurement units 604 will therefore be provided with reference to FIG. 6

[0169] Similarly to the representations shown in the other figures, the lines joining components shown in FIG. 4 represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0170] In addition to the two 2-pair measurement units 604, the measurement system 602 of FIG. 6 further comprises an extra switching stage comprising a further eight switches 504. Each of the switches 504 are 2x2 optical switches 504. The measurement system 602 of FIG. 6 has a total of sixteen inputs 606. Each input 606 of the measurement system 602 is coupled to an input of a switch 504, which has a first output coupled with a first of the 2-pair measurement units 604 and a second output coupled with a second of the 2-pair measurement units 604. Each input 606 can therefore be switched between either of the first of the 2-pair measurement units 604 or the second of the 2-pair measurement units 604. As was explained above, each of the 2-pair measurement units 604 has a plurality of inputs, each of which can be switched between one of the two measurement units 508 included in each 2-pair measurement unit 604. The switches 504 (which may be considered to form a switching network) are therefore operable to selectively direct a photon received at any of the inputs 606 of the measurement system 602 to any of the four measurement units 508.

[0171] In this way, the switches 504 may be operated to direct a first pair of photons received at two of the inputs 606 to a first of the measurement units 508, a second pair of photons received at two of the inputs 606 to a second of the measurement units 508, a third pair of photons received at two of the inputs 606 to a third of the measurement units 508 and / or a fourth pair of photons received at two of the inputs 606 to a fourth of the measurement units 508. The measurement system 602 is therefore operable to perform up to four Bell state measurements and create four quantum communication links in parallel.

[0172] In the example of FIG. 6, the measurement system 602 comprises sixteen inputs 606. In some examples, each input 606 may be coupled to a different quantum computing node 104 (not shown in FIG. 6). The measurement system 602 may therefore be operable to establish quantum communication links between one or more pairs of the sixteen quantum computing nodes 104.

[0173] Each optical path from an input 606 of the measurement system 602 of FIG. 6 to a measurement unit 508 passes through two optical switches 504. The measurement system 602 of FIG. 6 may therefore be described as including two switching stages. As can be seen in FIG. 6, some of the optical paths from an input 606 to a measurement unit 508 includes one or more crossing points at which two optical paths cross (for ease of illustration, the crossing points are not labelled with reference numerals in FIG. 6).

[0174] Similarly, to the arrangement shown in FIG. 5, in the arrangement shown in FIG. 6, half of the switches 504 are arranged on a first side (the left-hand side in FIG. 6) of the measurement units 508 (and 2-pair measurement units 604) and half of the switches 504 are arranged on a second side (the right-hand side of FIG. 6) of the measurement units 508 (and 2-pair measurement units 604). By positioning switches on both of two sides of the measurement units 508 and 2- pair measurement units 604 (e.g., the symmetrical arrangement shown in FIG. 6 in which half of the switches are positioned on each side), the number of crossing points 510 may be minimised.

[0175] As will be shown in further detail below, the arrangement of the measurement system 602 of FIG. 4 has been found to exhibit a lower number of switching stages and / or a lower number of crossing points than other comparable arrangements. Using an arrangement of the type shown in FIG. 6, the number of switching stages scales with log2(i / 4) and the number of optical crossings scales with i / 2-1 -log2(i / 2) for i >8, where i is the number of inputs 606.

[0176] The arrangement of the measurement system 602 of FIG. 6 may therefore exhibit lower optical losses than other comparable arrangements.

[0177] The measurement system 502 of FIG. 5 and the measurement system 602 of FIG. 6 each include switching networks comprising a plurality of 2 x 2 optical switches. More generally, according to examples disclosed herein, a switching network may comprise a plurality of M x M switches (i.e., M inputs and M outputs at each switch). According to at least some examples M may be greater than two. For example, a switching network according to at least some examples may comprise a plurality of 4 x 4 switches.

[0178] FIG. 7 is a schematic illustration of a measurement system 702 according to further examples of the present disclosure. The measurement system 702 of FIG. 7 is similar to and includes many of the same or corresponding components as the measurement system 602 described above with reference to FIG. 6. In particular, the measurement system 702 of FIG. 7 comprises two 2-pair measurement units 704. Each 2-pair measurement unit 704 is similar to the 2-pair measurement units 604 described above with reference to FIG. 6 m that it comprises two measurement units 508 and a plurality of switches 706. Each measurement unit 508 comprises a coupler 204 and a plurality of single photon detectors 210.

[0179] Corresponding components of the measurement units 508 depicted in FIG. 7 are provided with corresponding reference numerals to those used in FIG. 2, FIG. 5 and FIG. 6. Furthermore, corresponding components of the 2-pair measurement unit 704 depicted in FIG. 7 are provided with corresponding reference numerals to those used in FIG. 5 and FIG. 6. For ease of illustration, reference numerals may not be provided for each and every component of the measurement system 70202 shown in FIG. 7. The measurement units 508 of the measurement system 702 of FIG. 7 may include any of the features or components of the measurement system 202 described above with reference to FIG. 2 and / or the measurement units 508 described above with reference to FIG. 5 and / or FIG. 6. Furthermore, the 2-pair measurement unit 704 of the measurement system 702 of FIG. 7 may include any of the features or components of the measurement system 502 described above with reference to FIG. 5 and / or the 2-pair measurement unit 604 described above with reference to FIG. 6, except for the differences which are described below. No further detailed description of the operation of the measurement units 508 or 2-pair measurement units 704 will therefore be provided with reference to FIG. 7.

[0180] Similarly to the representations shown in the other figures, the lines joining components shown in FIG. 7 represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0181] The measurement system 70202 of FIG. 7 differs from the measurement system 602 of FIG. 6 in that the switching network comprises a plurality of 4 x 4 switches 706. In particular, the switching network of the measurement system 602 of FIG. 6 (which comprises a plurality of 2x2 switches) is replaced in the measurement system 702 of FIG. 7 with a plurality of 4 x 4 switches 706. In particular, both the outer switching stage, which is arranged to receive sixteen inputs 708 and the switches 706 which form part of the 2-pair measurement units 704 shown in FIG. 7 comprise 4x4 switches 706. Each 4x4 switch 706 in the example of FIG. 7 replaces two 2x2 switches 504 in the example of FIG. 6.

[0182] The 4x4 switches may comprise switches based on MMI couplers which may be referred to as MMI switches. As was explained above, when describing 2x2 MMI switches, an optical switch based on an MMI coupler may comprise an MMI coupler and at least one heater arranged to heat a portion of the MMI coupler to achieve optical switching. For example, a 2 x 2 MMI switch may comprise a 2 x 2 MMI coupler and at least one heater. Similarly, a 4 x 4 MMI switch may comprise a 4 x 4 MMI coupler and at least one heater arranged to heat at least a portion of the 4x4 MMI coupler to achieve optical switching. Additionally or alternatively a switch based on an MMI coupler may comprise an MMI coupler which is tuned using currentinjection tuning to achieve optical switching. Additionally or alternatively a switch based on an MMI coupler may comprise an MMI coupler which is tuned using the electro-optic effect to achieve optical switching. For example, at least one component (e.g., an electrode) may be provided which is operable to generate an electric field for inducing a change in the optical properties of the MMI coupler through the electro-optic effect.

[0183] In at least some examples, a switch based on an MMI coupler may comprise a plurality of MMI couplers. For example, a switch based on an MMI coupler may comprise a first MMI coupler and a second MMI coupler, where the outputs of the first MMI coupler are optically connected to the inputs of the second MMI coupler. Optical switching may be achieved by arranging phase shifters to apply controllable phase shifts along the optical paths between the outputs of the first MMI coupler and the second MMI coupler. The phase shifters may be controlled in order to achieve optical switching such that controllable switching may be achieved between the inputs of the first MMI coupler and the second MMI coupler. Examples of 4 x 4 MMI switches based on two 4x4 MMI couplers and controllable phase shifters are described, for example, in [4] and [5],

[0184] Each of the 4x4 switches 706 is arranged to receive four inputs. In particular, each of the 4 x 4 switches 706 forming the outer switching stage is arranged to receive four of the sixteen inputs 708 to the measurement system 702. Each 4x4 switch 706 of the outer switching stage has two outputs coupled with a first of the 2-pair measurement unit 704 and a further two outputs coupled with a second of the 2-pair measurement units 704. Each input 708 can therefore be switched between either of the first of the 2-pair measurement units 704 or the second of the 2-pair measurement units 704. As was explained above, each of the 2-pair measurement units 704 has inputs which can be switched between one of the two measurement units 508 included in each 2-pair measurement unit 704. In the measurement system 702 of FIG. 7, this is implemented by way of 4 x 4 switches 706 in the 2-pair measurement units 704 (which replace the 2x2 switch 504 included in the 2-pair measurement units 604 shown in FIG. 6). The 4x4 switches 706 (which may be considered to form a switching network) are therefore operable to selectively direct a photon received at any of the inputs 708 of the measurement system 702 to any of the four measurement units 508.

[0185] Similarly to the measurement system 602 of FIG. 6, the switching network may be operated to direct up to four pairs of photons received at different pairs of the inputs 708 to different measurement units 508. The measurement system 702 is therefore operable to perform up to four Bell state measurements and create up to four quantum communication links in parallel.

[0186] When compared to the measurement system 602 of FIG. 6, the measurement system 702 of FIG. 7 includes the same number of inputs 708 and is operable to perform the same number of Bell state measurements in parallel. Furthermore, the measurement systems 602, 702 of FIG. 6 and FIG. 7 include the same number of switching stages (two) for each optical path. However, the 4x4 switches 706 included in the measurement system 702 FIG. 7 reduces the total number of crossing points of optical paths (when compared to the measurement system 602 of FIG. 6). For example, using an arrangement of the type shown in FIG. 7, the number of crossing points may scale with 20 / 8-1), where i is the number of inputs 708 (as compared to scaling with i / 2-1 -log20 / 2) according to an arrangement of the type shown in FIG. 6). This reduction in the number of crossing points serves to reduce the optical losses along the optical paths.

[0187] When compared to the measurement system 602 of FIG. 6 (which includes 2x2 switches), the use of 4 x 4 switches in the measurement system 702 of FIG. 7 further increases the combinations of inputs 708 at which a pair of photons can arrive and a Bell state measurement of the photons performed. As was explained above with reference to FIG. 5, a Bell state measurement cannot be performed on two photons arriving at inputs of the same switch 504. This is because each 2x2 switch 504 only has a single output going to each measurement unit 508. In the arrangements shown in FIG. 5 and FIG. 6, a 2 x 2 switch 504 cannot therefore be operated to simultaneously direct two photons arriving at two inputs of the same switch 504 to the same measurement unit 508 using different outputs of the 2x2 switch (since each 2 x 2 as a single output directed to each measurement unit 508). In contrast to this limitation, in the arrangement shown in FIG. 7, each 4x4 switch 704 has two outputs directed to each measurement unit 508. A 4 x 4 switch 706 can therefore be operated to direct two photons arriving at two of its inputs to the same measurement unit 508 (along the two outputs which are directed to the same measurement unit 508). In this way the switches 706 may be operated to facilitate a Bell state measurement between photons arriving at any two of the inputs 708 (including inputs 708 which are inputs to the same switch 706).

[0188] Similarly to the arrangements of FIG. 5 and FIG. 6, in the arrangement shown m FIG. 7, half of the switches 706s are arranged on a first side (the left-hand side in FIG. 7) of the measurement units 508 (and 2-pair measurement units 704) and half of the switches 706s are arranged on a second side (the right-hand side of FIG. 7) of the measurement units 508 (and 2-pair measurement units 704). By positioning switches on both of two sides of the measurement units 508 and 2-pair measurement units 704 (e.g., the symmetrical arrangement shown in FIG. 7 in which half of the switches are positioned on each side), the number of crossing points 510 may be minimised.

[0189] As was explained above with reference to FIG. 7, in at least some examples, at least one 4x4 switch may be used in a measurement system 702. The use of 4 x 4 switches may enable a reduction in a number of crossing points of optical paths in a measurement system 702. As was further described above, in at least some examples, a 4 x 4 switch may be realised as a 4 x 4 MMI switch (a 4 x 4 switch based on a 4 x 4 MMI coupler). A 4 x 4 MMI switch may comprise a single MMI coupler or may comprise a plurality of MMI couplers (e.g., with controllable phase shifters situated in the optical paths between MMI couplers). Additionally or alternatively, in at least some examples, a 4 x 4 switch may comprise a plurality of 2 x 2 switches arranged to provide the functionality of a 4 x 4 switch.

[0190] FIG. 8 is a schematic illustration of an example of a 4 x 4 switch 706 which is realised as an arrangement of a plurality of 2 x 2 switches 802. One or more of the 4x4 switches 706 of the measurement system 702 shown in FIG. 7 may, in some examples, be realised as an arrangement of 2 x 2 switches 802 as shown in FIG. 8. The 2x2 switches 802 may comprise 2 x 2 MMI switches or any other suitable form of 2 x 2 optical switch. The 4x4 switch 706 shown in FIG. 7 comprises a first switching stage 808, a second switching stage 810 and a third switching stage 812. Each switching stage 808, 810, 812 comprises two 2x2 switches. The inputs to the two switches 802 in the first switching stage 808 form four inputs 804 to the 4x4 switch 706.

[0191] Similarly to the representations shown in the other figures, the lines joining components shown in FIG. 8 represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0192] Each switch 802 in the first switching stage 808 comprises two outputs which are coupled to inputs of different switches 802 in the second switching stage 810. Similarly, each switch 802 in the second switching stage 810 comprises two outputs which are coupled to inputs of different switches 802 in the third switching stage 812. The outputs of the two switches 802 in the third switching stage 812 form four outputs 806 of the 4x4 switch 706.

[0193] The arrangement of 2 x 2 switches 802 shown in FIG. 8 allows each of the inputs 804 to be connected to any of the outputs 806. Furthermore, the arrangement of 2 x 2 switches 802 shown in FIG. 8 is a non-blocking arrangement such that in the presence of an established connection between an input 804 and an output 806, each unused input 804 may be directed to any of the unused outputs 806 whilst maintaining the established connection. The arrangement of switches 802 shown in FIG. 8 forms a 4 x 4 Benes network. The principles of a Benes network topology is described in more detail for example in [6], A 4 x 4 switch 706 of the form shown in FIG. 8 and comprising a Benes network may be referred to as a 4 x 4 Benes switch.

[0194] As was explained above with reference to FIG. 7, the use of 4 x 4 switches 706 (when compared to the use of 2 x 2 switches) in a measurement system 702 may reduce the total number of crossing points in optical paths which may serve to reduce the optical losses along the optical paths. As can be seen in FIG. 8 when a 4 x 4 switch 706 is realised as a Benes network of 2 x 2 switches, the 4x4 switch 706 itself includes a crossing point along at least some of the optical paths between the inputs 804 and the outputs 806. These crossing points may serve as a source of optical loss along at least some of the optical paths (when compared, for example, to using 4 x 4 MMI switches). However, as will be shown in detail below, the use of 4 x 4 switches realised as a Benes network of 2 x 2 switches in a measurement system 702 may still provide a reduction in optical losses when compared, for example, to the use of 2 x 2 switches (e.g., as shown in the measurement system 602 of FIG. 6). Furthermore, the use of 4 x 4 switches realised as a Benes network of 2 x 2 switches also allows a Bell state measurement to be performed between a pair of photons arriving at any of the inputs 708 (including inputs of the same 4x4 switch) as was described with reference to FIG. 7.

[0195] In further examples, M x M switches may be used, where M is greater than 4. Such M x M switches may be realised as MMI switches comprising at least one MMI coupler comprising greater than 4 inputs and greater than 4 outputs. Additionally or alternatively, such M x M switches may be realised as a Benes network of switches.

[0196] Several examples of measurement systems 202, 502, 602, 702 have been described above with reference to FIG. 2 - FIG. 8. Each of the measurement systems 202, 502, 602, 702 include at least a coupler 204 comprising N input ports and N output ports, where N is greater than two (N=4 in each of the depicted examples) and a Bell state measurement apparatus comprising a plurality of single photon detectors 210 arranged to receive and measure photons output from a plurality of the N output ports. The Bell state measurement apparatus is configured to project measurements made by the single photon detectors 210 onto a Bell state so as to perform a Bell state measurement.

[0197] In the example measurement system 202 of FIG. 2, a single coupler 204 is included and the measurement system 202 is capable of performing a single Bell state measurement of a single pair of photons at a time. The number of Bell state measurements of pairs of photons that a measurement system can perform in parallel may be referred to herein as the number of input modes of the measurement system. The measurement system 202 of FIG. 2 has one input mode.

[0198] The measurement system 202 of FIG. 2 forms a measurement unit 508, which can be used as a building block to produce larger measurement systems capable of performing a plurality of Bell state measurements between a plurality of pairs of photons in parallel and / or to receive photons from a larger number of inputs.

[0199] For example, the measurement system 502 of FIG. 5 includes two measurement units 508 and a switching network operable to selectively direct a photon received at any of a plurality of inputs to either of the measurement units 508. The two measurement units 508 can perform Bell state measurements of different pairs of photons in parallel and thus the measurement system 502 of FIG. 5 is operable to perform two Bell state measurements of two different pairs of photons in parallel. The measurement system 502 of FIG. 5 therefore has two input modes.

[0200] The measurement systems 602, 702 shown in FIG. 6 and FIG. 7 respectively each include four measurement units 508 and a two stage switching network which is operable to selectively direct a photon received at any of a plurality of inputs to any of the four measurement units 508. The measurement units 508 can perform Bell state measurements of different pairs of photons in parallel and thus the measurement systems 602, 702 of FIG. 6 and FIG. 7 are each operable to perform four Bell state measurements of four different pairs of photons in parallel. The measurement systems 602, 702 of FIG. 6 and FIG. 7, therefore each have four input modes.

[0201] The principles and features described above with reference to the measurement systems 202, 502, 602, 702 described above with reference to FIG. 2 - FIG. 8 may be further scaled up to include larger numbers of measurement units 508 and / or larger numbers of inputs. The principles and features disclosed herein may therefore be used to perform efficient and low loss Bell state measurements of pairs of photons originating from any number of different quantum computing nodes 104 in a quantum computing system 102. The number of measurement units 508 (and thus the number of photon pairs for which Bell state measurements can be performed in parallel) and / or the number of inputs to a measurement system may be scaled to any suitable number according to the number of quantum computing nodes 104 included in a quantum computing system 102. Scaling of a measurement system to be able to perform Bell state measurements of a larger number of different pairs of photons in parallel may be described as scaling the number of input modes of a measurement system.

[0202] FIG. 9 is a graph illustrating a total optical loss in measurement systems according to some of the principles disclosed herein as a function of the size of the measurement system. The total optical loss in decibels (dB) is plotted on the y-axis of the graph of FIG. 9 versus a number of input modes of the measurement system on the x-axis of the graph of FIG. 9. As was explained above, the total number of input modes as shown on the x-axis of the graph of FIG. 9 corresponds to the number of different pairs of photons of which Bell state measurements can be made in parallel. For example, according to the principles disclosed herein, the number of input modes may equivalently be considered as the number of measurement units 508 included in a measurement system. The total optical losses shown in FIG. 9 are calculated on the basis that each switch which is traversed by an optical path results in an optical loss of 0.5 dB and each crossing of optical paths which is traversed by an optical path results in an optical loss of 0.1 dB.

[0203] Results corresponding to four different measurement system arrangement types are plotted in FIG. 9. Three of the different measurement system arrangement types (shown with lines 904, 906 and 908) correspond to measurement system arrangements according to examples disclosed herein. These results are benchmarked against results shown with a line labelled 902 corresponding to the use of a Benes switching network. For each number of input modes, which is plotted in FIG. 9, the results 902 shown for a Benes switching network correspond to a Benes switching network arranged to provide fully-connected (each input is connectable to each output) non-blocking switching between a plurality of inputs and a plurality of Bell state measurement apparatus (each with two inputs). The number of Bell state measurement apparatus (each with two inputs) which the Benes switching network is arranged to provide connectivity to, corresponds to the number of nodes plotted on the x-axis of FIG. 9. A Benes switching network with X input modes is therefore operable to perform X Bell state measurements of X pairs of photons in parallel.

[0204] The results labelled 904 in FIG. 9 are calculated for the use of measurement systems 602 of the type shown in FIG. 6 in which each switching stage comprises 2x2 optical switches. The results labelled 906 in FIG. 9 are calculated for the use of measurement systems 702 of the type shown in FIG. 7 in which 4x4 switches are used in the switching stages and in which the 4x4 switches are realised as 4 x 4 MMI switches. The results labelled 908 in FIG. 9 are calculated for the use of measurement systems 702 of the type shown in FIG. 7 in which 4x4 switches are used in the switching stages and in which the 4x4 switches are realised as a 4 x 4 Benes switching network as was described above with reference to FIG. 8. As was explained above, for each of the measurement system arrangement types (2x2 switches 904, 4x4 MMI switches 906 and 4x4 Benes switches 908) according to examples disclosed herein, the number of input modes corresponds with the number of measurement units 508 included in the measurement system and thus the number of Bell state measurements of different pairs of photons which can be performed in parallel.

[0205] For each of the measurement system arrangement types (2x2 switches 904, 4x4 MMI switches 906 and 4x4 Benes switches 908) according to examples disclosed herein the measurement systems also provide fully-connected and non-blocking switching of all inputs to each measurement unit 508 (with the exception of two inputs of the same 2x2 switch in the 2 x 2 switching 904 implementation). For the same number of input modes, each of the measurement system arrangement types (2x2 switches 904, 4x4 MMI switches 906 and 4x4 Benes switches 908) according to examples disclosed herein, therefore provide comparable functionality to the equivalent Benes switching network 902 having the same number of input modes.

[0206] As can be seen in FIG. 9 the total optical loss for each of the measurement system arrangement types (2x2 switches 904, 4x4 MMI switches 906 and 4x4 Benes switches 908) according to examples disclosed herein are significantly less than the total optical loss for an equivalent Benes switching network 902. The lowest optical losses (for all number of input modes) are seen for measurement systems comprising 4x4 MMI switches 906. At low numbers of input modes (less than about 45) lower optical losses are seen for measurement systems comprising 2x2 switches 904 than measurement systems comprising 4x4 Benes switches 908. However, as the number of input modes increases (above about 45) this trend is reversed and measurement systems comprising 4x4 Benes switches 908 exhibit lower total losses than measurement systems comprising 2x2 switches 904.

[0207] As was explained above, the use of a 4 x 4 coupler 204 in a measurement system (as opposed to, for example, using a 2 x 2 coupler) means that up to two times the number of single photon detectors 210 may be used to detect the same number of photon pairs. For example, if a single photon detector 210 is arranged at each output port of each 4x4 coupler 204 included in a measurement system, then twice the number of single photon detectors 210 may be used, when compared to the use of 2 x 2 couplers to detect the same number of pairs of photons. As was explained above, the results shown in FIG. 8 were obtained by including additional optical loss associated with reducing the number of single photon detectors 210 used per input mode such that the number of detectors used per input mode for each measurement system type 904, 906, 908 is the same as for a comparable Benes switching network 902 A comparison which accounts for the difference in number of detectors used per input mode may be made by plotting the optical loss for a Benes switching network 902 having x input modes against the optical loss for the other types of measurement system 904, 906, 908 having 2x input modes.

[0208] FIG. 10 is a further graph illustrating total optical loss in different measurement systems as a function of the number of input modes. FIG. 10 is similar to FIG. 9 and results corresponding to corresponding measurement system types are denoted with the same reference numerals in FIG. 10 and FIG. 9. For example, the results labelled 902 correspond to the use of a Benes switching network as described above with reference to FIG. 9. The results labelled 904 in FIG. 10 correspond to the use of measurement systems including 2x2 switches as described above with reference to FIG. 9. The results labelled 906 in FIG. 10 correspond to the use of measurement systems including 4x4 MMI switches as described above with reference to FIG. 9. The results labelled 908 in FIG. 10 correspond to the use of measurement systems including 4x4 Benes switches 908 as also described above with reference to FIG. 9. The results labelled 904, 906, 908 in FIG. 10 differ from those labelled with the same reference numerals in FIG. 9 in that in FIG. 10 the optical losses associated with the measurement system types labelled 904, 906, 908 are plotted at a position on the x-axis corresponding to half of the input modes actually used in the measurement system. That is, the optical losses for measurement system types labelled 904, 906, 908 and having 2x input modes are plotted on the same position on the x-axis as a Benes switching network 902 having x input modes. This may equivalently be thought of as comparing the optical losses of a Benes switching network with the optical losses of a measurement system of double the size (double the number of input modes) of the Benes switching network.

[0209] As can be seen by comparing FIG. 10 with FIG. 9, even when doubling the number of input modes in making the comparison to a Benes switching network, each of the measurement system types 904, 906, 908 generally exhibit lower optical loss than an equivalent Benes switching network 902. This is especially true as the number of input modes scales up. For example, for more than about twenty input modes all of the measurement system types 904, 906, 908 exhibit lower optical loss than the comparable Benes switching network 902. Furthermore, the use of a measurement system comprising 4x4 MMI switches 906, and 2x2 MMI switches 904 exhibit lower optical loss than the comparable Benes switching network 902 even for measurement systems comprising less than twenty input modes.

[0210] Various examples of measurement systems have been described herein in which a measurement system 202 and / or measurement unit 508 comprises a single optical coupler 204. For example, in the example shown in FIG. 2 a measurement system 106 system comprises an optical coupler 204 having four input ports 206 and four output ports 208 (a 4 x 4 coupler), single photon detectors 210 and a measurement analyser 212. These components form a measurement unit 508, and a plurality of measurement units 508 are included in the example measurement systems 502, 602, 702 of each of FIG. 5, FIG. 6 and FIG. 7. In these examples, each measurement unit 508 (and the measurement system 202 of FIG. 2) can perform a Bell state measurement of a pair of photons arriving at two of the input ports 206. In each of these examples, each measurement unit 508 (and the measurement system 202 of FIG. 2) may be described as comprising an optical circuit having N input ports and N output ports (where N is four in the depicted examples). In the examples, of FIG. 2, FIG. 5, FIG. 6 and FIG. 7 the optical circuit comprises just the coupler 204 also having N input ports and N output ports. In these examples, the N input ports of the optical circuit correspond with the N input ports of the coupler 204 and the N output ports of the optical circuit correspond with the N output ports of the coupler 204.

[0211] In some examples, a measurement unit may be arranged so as to be able to perform independent Bell state measurements of two pairs of photons, where each photon of the two photon pairs arrives at different input ports of the measurement unit. FIG. 11 is a schematic illustration of a measurement unit 1112 comprising four input port 1108. As will be described in further detail below, the measurement unit 1112 can perform two independent Bell state measurements of two pairs of photons (total of four photons) where each photon arrives at a different input port 1108 of the measurement unit 1112. For example, the measurement unit 1112 may perform independent Bell state measurements of two pairs of photons arriving at the input ports 1108 at substantially the same time.

[0212] The measurement unit 1112 comprises an optical circuit 1102 comprising a 4 x 4 optical coupler 1104 and a 2 x 2 optical couplers 1106. The optical circuit 1102 comprises four input ports 1108 and four output ports 1110. The measurement unit 1112 further comprises four single photon detectors 1114. In the example shown in FIG. 11 the measurement unit 1112 further comprises a measurement analyser 1116. The lines representing the input ports 1108, output ports 1110 and lines joining the various optical components depicted in FIG. 11 represent optical waveguides arranged to support propagation of photons between the components which are joined by the lines.

[0213] The 4x4 optical coupler 1104 may comprise any of the features described above with reference to the coupler 204 described with reference to FIG. 2. For example, the 4x4 optical coupler 1104 may comprise an MMI coupler. The 4x4 optical coupler 1104 comprises four input ports and four output ports as shown in FIG. 11. Both of the 2 x 2 couplers 1106 may comprise any suitable passive optical couplers. For example, the 2x2 couplers 1106 may comprise MMI optical couplers. The 2x2 couplers 1106 each comprise two input ports and two output ports as shown in FIG. 11.

[0214] There are a number of different optical paths that can be traversed by a photon entering any of the input ports 1108 of the optical circuit 1102. Each possible optical path links to one of the four output ports 1110 of the optical circuit 1102. The single photon detectors 1114 are arranged to measure photons output from the output port 1110 of the optical circuit 1102. The single photon detectors 1114 may comprise any of the features described above with reference to the single photon detectors 210 described with reference to FIG. 2. For example, the single photon detectors 1114 may be of a type having a relatively short dead-time after detection of a first photon and before a second subsequent photon can be detected, so as to allow for detection of photons in multiple time bins (e.g., in both an early and a late time bin).

[0215] The optical circuit 1102 may perform a similar role to the 4x4 coupler 204 used in the measurement system 202 of FIG. 2 and the measurement units 508 of the examples of FIG. 5, FIG. 6 and FIG. 7. In particular, it has been found that for a pair of photons arriving at known input ports 1108 of the optical circuit 1102, the output ports 1110 at which photons are detected and time bins in which photons are detected can be used to project measurements made by the single photon detectors 1114 onto a Bell state. The measurement analyser 1116 may be communicatively coupled to the single photon detectors 1114 and configured to receive data from the single photon detectors 1114 indicative of the measurements made by the single photon detectors 1114. The data may be in any suitable form and may, for example, comprise digital data and / or analogue signals. For example, the data may comprise signal pulses generated by a single photon detector 1114 at a time at which the single photon detector 1114 has detected a photon.

[0216] The measurement analyser 1116 may be configured to determine a Bell state of two photons based on the measurements made by the single photon detectors 1114. In particular, the measurement analyser 1116 may be configured to determine a Bell state of two photons based on which of the single photon detectors 1114 detect the photons (and hence the output ports 1110 at which the photons are measured), the time bins in which the photons are detected and the input ports 1108 at which the photons arrive.

[0217] The measurement unit 1112 of FIG. 11 based on the optical circuit 1102 has an additional advantage in that two independent Bell state measurements of two pairs of photons arriving at different input ports input port 1108 can be performed in parallel.

[0218] The optical circuit 1102 used in the measurement unit 1112 of FIG. 11 is one example of an optical circuit 1102 which can be used to facilitate Bell state measurements of two pairs of photons in parallel. In other examples, different arrangements of optical circuits may be used. Two further examples are shown in FIG. 12 and FIG. 13.

[0219] FIG. 12 is a schematic illustration of a measurement unit 1210 comprising an optical circuit 1208. Similarly to the optical circuit 1102 shown in FIG. 11, the optical circuit 1208 of FIG. 12 comprises four input port 1108 and four output ports 1110. The optical circuit 1208 of FIG. 12 also includes a 4 x 4 optical coupler 1104 and a plurality of 2 x 2 optical couplers 1106 arranged to provide a number of different optical paths that can be traversed by a photon entering any of the input ports 1108 of the optical circuit 1208. Each possible optical path links to one of the four output ports 1110 of the optical circuit 1208. The optical circuit 1208 of FIG. 12 further includes phase shifters 1206. The phase shifters 1206 are configured to bring about a phase shift in photons propagating along the optical paths which pass through the phase shifters 1206. In the example, of FIG. 12 each of the phase shifters 1206 is configured to bring about a phase shift of 7t / 4.

[0220] Similarly, to the measurement unit 1112 of FIG. 11, the measurement unit 1210 of FIG. 12 also includes four single photon detectors 1114 and a measurement analyser 1116. These components may include any of the features described above with reference to any of the earlier figures.

[0221] The optical circuit 1208 of the example of FIG. 12 has similar functionality to the optical circuit 1102 of FIG. 11 and it has been similarly found that for a pair of photons arriving at known input ports 1108 of the optical circuit 1208, the output ports 1110 at which photons are detected and time bins in which photons are detected can be used to project measurements made by the single photon detectors 1114 onto a Bell state.

[0222] FIG. 13 is a schematic illustration of a further example of a measurement unit 1306 comprising an optical circuit 1302. Similarly to the optical circuit 1102 shown in FIG. 11 and the optical circuit 1208 shown in FIG. 12, the optical circuit 1302 of FIG. 13 comprises four input ports 1108 and four output ports 1110. The optical circuit 1304 of FIG. 13 also includes a 4x4 optical coupler 1104 and a plurality of 2 x 2 optical couplers 1106 arranged to provide a number of different optical paths that can be traversed by a photon entering any of the input ports 1108 of the optical circuit 1302. Each possible optical path links to one of the four output ports 1110 of the optical circuit 1302. The optical circuit 1302 of FIG. 12 further includes phase shifters 1304. The phase shifters 1304 are configured to bring about a phase shift in photons propagating along the optical paths which pass through the phase shifters 1304. In the example of FIG. 13, each of the phase shifters 1304 is configured to bring about a phase shift of n / 2.

[0223] Similarly to the measurement unit 1112 of FIG. 11 and the measurement unit 1210 of FIG. 12, the measurement unit 1306 of FIG. 13 also includes four single photon detectors 1114 and a measurement analyser 1116. These components may include any of the features described above with reference to any of the earlier figures.

[0224] The optical circuit 1302 of the example of FIG. 13 has similar functionality to the optical circuit 1102 of FIG. 11 and the optical circuit 1208 of FIG. 12 and it has been similarly found that for a pair of photons arriving at known input ports 1108 of the optical circuit 1302, the output ports 1110 at which photons are detected and time bins in which photons are detected can be used to project measurements made by the single photon detectors 1114 onto a Bell state.

[0225] As was explained above, the optical circuits 1102, 1208, 1302 of the examples of FIG. 11, FIG. 12 and FIG. 13 are arranged such that measurements of photons made by the single photon detectors 1114 can be projected on to a Bell state. FIG. 14 is a table representing all of the measurement combinations which correspond with Bell state measurements of two photon pairs arriving at input ports 1108 of any of the optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13. In each of FIG. 11, FIG. 12 and FIG. 13, the input ports 1108 are denoted with numbers 1,2,3 and 4. In order to perform Bell state measurements of two pairs of photons in parallel, a first photon pair is generated to arrive at input ports 1 and 4 and a second photon pair is generated to arrive at input ports 2 and 3. As was described above, each photon may be generated to as to arrive at an input port 1108 in a superposition of an early time bin and a late time bin.

[0226] The input port pairs are shown in the two left-most columns of the table of FIG. 14. In the first column of the table of FIG. 14 the input port combinations of a first pair of photons are shown and in the second column of FIG. 14 the input port combination of a second pair of photons are shown. The input ports 1108 are denoted with numbers 1, 2, 3 and 4 corresponding to the input port 1108 numbering shown in FIG. 11, FIG. 12 and FIG. 13. In particular, the data shown in the table of FIG. 14 corresponds to a first photon pair arriving at input ports 1 and 4 and a second photon pair arriving at input ports 2 and 3. Each photon in both the first photon pair and the second photon pair is considered to arrive at an input port in a superposition of an early time bin and a late time bin.

[0227] The different output port pairs at which photon measurements (at which a pair of photons might be detected) are shown in the upper-most row of the table of FIG. 14. The output ports 1110 are denoted with numbers 1, 2, 3 and 4 corresponding to the output port 1110 numbering shown in FIG. 11, FIG. 12, and FIG. 13. The crosses marked in the table of FIG. 14 indicate the output port measurements which correspond with a Bell state for the first or second pair of photons. Two crosses in a single box in the table of FIG. 14 indicate measurements of two photons in the different time bins (as denoted by tO and tl in the headings in the second row of the table of FIG. 14). Measurements shown in a black box (with white crosses) in FIG. 14 correspond to a measurement of the first pair of photons in the >Bell state and measurements shown in light-grey boxes (with black crosses) in FIG. 14 correspond to a measurement of the second pair of photons in the >Bell state (as is indicated in the key in FIG. 14).

[0228] As shown m FIG. 14 if a first pair of photons arrive at input ports 1 and 4 and a second pair of photons arrive at input ports 2 and 3, then a photon detection in the early time bin (tO) and the late time bin (tl) at output port 1 corresponds to a projection of the first photon pair into the Bell state. Similarly, a photon detection in the early time bin (tO) and the late time bin (tl) at output port 2 corresponds to a projection of the first photon pair into the - >Bell state. A photon detection in the early time bin (tO) at one of output port 1 or output port 2 and in the late time bin (tl) at the other of output port 1 or output port 2 also corresponds to a projection of the first photon pair into the >Bell state. A photon detection in the early time bin (tO) and the late time bin (tl) at output port 3 or output port 4 corresponds to a projection of the second photon pair into the Bell state. A photon detection in the early time bin (tO) at one of output port 3 or output port 4 and in the late time bin (tl) at the other of output port 3 or output port 4 also corresponds to a projection of the second photon pair into the >Bell state.

[0229] As is demonstrated by the results shown in FIG. 14, the optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13 can be used to perform independent Bell state measurements of two photon pairs in parallel. In particular, this approach allows two independent Bell state measurements of two photon pairs in parallel with the use of four single photon detectors 1114. By way of comparison, the measurement system 202 of FIG. 2 and the measurement units 508 used in the examples of FIG. 5, FIG. 6 and FIG. 7 allow for a single Bell state measurement of a single photon pair using four single photon detectors 210 and two independent Bell state measurements of two photon pairs in parallel (using two measurement units 508) uses eight single photon detectors 210. The optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13 may therefore reduce the number of single photon detectors used to perform two Bell state measurements in parallel (when compared to the measurement system 202 of FIG. 2 and the measurement units 508 used in the examples of FIG. 5, FIG. 6 and FIG. 7).

[0230] Whilst the optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13 include some 2x2 couplers, the use of a 4 x 4 coupler in the optical circuits 1102, 1208, 1302 provides similar advantages to those described above with reference to the use of a 4 x 4 coupler 204 in the examples of FIG. 2, FIG. 5, FIG. 6 and FIG. 7. For example, the use of a 4 x 4 coupler 1104 optical circuits 1102, 1208, 1302 provides optical mixing between photon paths which allows measurements made by the single photon detectors 1114 to be projected onto a Bell state for photons arriving at the input ports 1108.

[0231] The optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13 may be considered to mimic the behaviour of two independent 2x2 couplers (which may function as beam splitters) with switching stages arranged to direct different photon pairs to different 2x2 couplers, such that each coupler is used to perform two Bell state measurements in parallel. When compared to such an arrangement, the optical circuits 1102, 1208, 1302 of FIG. 11, FIG. 12 and FIG. 13 which include a 4 x 4 optical coupler allow for less switching stages and crossing points thereby reducing optical loss.

[0232] In some examples, a plurality of measurement units 1118, 1210, 1306 as described with reference to FIG. 11, FIG. 12 and / or FIG. 13 may be included in a larger measurement system. For example, in any of the example measurement systems 502, 602, 702 of FIG. 5, FIG. 6 or FIG. 7, one or more measurement units 1118, 1210, 1306 as described with reference to any of FIG. 11, FIG. 12 and / or FIG. 13 may be included in place of a measurement unit 508. Since each measurement unit 1118, 1210, 1306 can perform independent Bell state measurements of two photon pairs (as opposed to one photon pair for the measurement units 508), the number of photon pairs for which independent Bell state measurements can be performed in parallel would be doubled for each measurement system 502, 602, 702 by using measurement units 1118, 1210, 1306 as described with reference to any of FIG. 11, FIG. 12 and / or FIG. 13. In this way the number of switching stages and optical crossings needed for each photon pair to be entangled can be decreased.

[0233] Whilst measurement units 1118, 1210, 1306 such as those shown in FIG. 11, FIG. 12 and / or FIG. 13 may be used in a larger measurement system (such as those shown in FIG. 5, FIG. 6 or FIG. 7), as was described above, the Bell state measurements made by the measurement units 1118, 1210, 1306 of FIG. 11, FIG. 12 and FIG. 13 rely on photon pairs arriving at specific input ports of the measurement units measurement units 1118, 1210, 1306. In particular, a first photon pair arrives at input ports 1 and 4 and a second photon pair arrives at input ports 2 and 3. In contrast to this, the measurement units 508 included in the measurement systems 502, 602, 702 of FIG. 5, FIG. 6 or FIG. 7 can receive any two photons at any two of the input ports to perform a Bell state measurement. This difference can be accounted for by including a single extra switching stage (for each measurement unit) in a measurement system which utilises measurement unit 1118, 1210, 1306 of FIG. 11, FIG. 12 and / or FIG. 13.

[0234] FIG. 15 is a schematic illustration of a measurement module 1502 according to examples disclosed herein. The measurement module 1502 comprises a measurement unit 1512 and a switching stage. In the example of FIG. 15 the switching stage comprises two 2x2 optical switches 1506, where the outputs of each switch 1506 form two of the inputs to the measurement unit 1512. The measurement unit 1512 may comprise any of the measurement units 1112, 1210, 1302 described above with reference to FIG. 11, FIG. 12 and FIG. 13. In particular, the measurement unit 1512 comprises an optical circuit 1508 and single photon detectors 1510. The optical circuit 1508 may comprise an optical circuit 1102, 1210, 1302 as described above with reference to any of FIG. 11, FIG. 12 or FIG. 13. The switching stage (comprising two 2x2 switches 1506) may be operated to direct photons such that a first photon pair for which a first Bell state measurement is to be performed is directed such that one photon of the first photon pair arrives at input port 1 of the measurement unit 1512 and the other photon of the first photon pair arrives at input port 4 of the measurement unit 1512. The switching stage may be further operated to direct photons such that a second photon pair for which a second Bell state measurement is to be performed is directed such that one photon of the second photon pair arrives at input port 2 of the measurement unit 1512 and the other photon of the second photon pair arrives at input port 3 of the measurement unit 1512. As was described above, this allows a first Bell state measurement of the first photon pair and a second Bell state measurement of the second photon pair to be performed in parallel with each other by the measurement unit 1512.

[0235] A measurement module 1502 of the form shown in FIG. 15 may be used to replace one or more measurement units 508 in any of the examples shown in FIG. 5, FIG. 6 or FIG. 7. In such example measurement systems up to double the number of Bell state measurements of photon pairs could be performed in parallel. FIG. 16 is a schematic illustration of an example, measurement system 1602 including two measurement modules 1502 as described above with reference to FIG. 15. The example measurement system 1602 of FIG. 16 is similar to the measurement system 502 of FIG. 5 except that the measurement units 508 of FIG. 5 are replaced with measurement modules 1502 as described with reference to FIG. 15. In particular, the measurement system 1602 of FIG. 16 comprises two measurement modules 1502 and a plurality of 2 x 2 optical switches to appropriately direct photons to one of the measurement modules 1502. Since each measurement module 1502 can perform two Bell state measurements of two photon pairs in parallel, the measurement system 1602 of FIG. 16 may be operable to perform four Bell state measurements of four photon pairs in parallel. In contrast the measurement system 502 of FIG. 5 can perform two Bell state measurements of two photon pairs in parallel (one Bell state measurement of one photon pair for each measurement unit 508). The use of measurement modules 1502 can therefore double the number of Bell state measurements that can be performed in parallel.

[0236] When compared to the measurement system 502 of FIG. 5, the measurement system 1602 of FIG. 16 includes an additional modification of the optical paths between the optical switches switches 1606 and the lower most 1604 shown in FIG. 16. This modification allows photons arriving at any input of the measurement system 1602 to be routed to an appropriate input of the measurement units 1512 for performing upto four Bell state measurements in parallel.

[0237] The example of FIG. 16 is provided purely as one illustrative example. Many other examples of measurement systems including measurement modules 1502 of the type described with reference to FIG. 15 may be used. For example, in the example measurement systems of FIG. 6 or FIG. 7, any of the measurement units 508 may be replaced with a measurement module 1502 of the type described with reference to FIG. 15.

[0238] FIG. 17 is a further graph illustrating a total optical loss in measurement system according to some of the principles disclosed herein as a function of the size of the measurement system. Similarly to FIG. 9 and FIG. 10, the total optical loss in decibels (dB) is plotted on the y-axis of the graph of FIG. 17 versus a number of input modes of the measurement system on the x-axis of the graph of FIG. 17. As was explained above, the total number of input modes as shown on the x-axis of the graph of FIG. 17 corresponds to the number of different pairs of photons of which Bell state measurements can be made in parallel. The total optical losses shown in FIG. 17 are calculated on the basis that each switch which is traversed by an optical path results in an optical loss of 0.5 dB and each crossing of optical paths which is traversed by an optical path results in an optical loss of 0.1 dB.

[0239] Results corresponding to two different measurement system arrangement types are plotted in FIG. 17. The results labelled 1702 in FIG. 17 are calculated for the use of measurement systems including measurement units 508 of the type used in the measurement systems 502, 602, 702 of FIG. 5, FIG. 6 and FIG. 7. That is, the results labelled 1702 in FIG. 17 are calculated for the use of measurement systems including measurement units 508 which can each perform a Bell state measurement of a single pair of photons. The results labelled 1704 in FIG. 17 are calculated for the use of measurement systems which include measurement modules 1502 of the type described with reference to FIG. 15. That is, the results labelled 1704 in FIG. 17 are calculated for the use of measurement systems including measurement modules 1502 which can each perform two independent Bell state measurements of two pairs of photons in parallel.

[0240] As can be seen in FIG. 17 the total optical losses corresponding to the use of measurement modules 1502 1502 as described with reference to FIG. 15 are less per input mode than the total optical losses corresponding to the use of measurement units 508 as described with reference to FIG. 2, FIG. 5, FIG. 6 and FIG. 7. This is because, as described above, the use of measurement modules 1502 as described with reference to FIG. 15 allows for the measurement of double the number of photon pairs (input modes) for a similar number of switching stages (when compared to the use of measurement units 508 as described with reference to FIG. 2, FIG. 5, FIG. 6 and FIG. 7).

[0241] A number of different examples of measurement systems and measurement units have been described and depicted herein. In each of the examples, a measurement system includes an optical circuit comprises N input ports and N output ports where N is greater than two and where the optical circuit comprises an optical coupler having N input ports and N output ports. In the examples, of FIG. 2, FIG. 5, FIG. 6 and FIG. 7 the optical circuit comprises just the optical coupler 204 such that the input and output ports of the optical circuit correspond directly with the input and output ports of the optical coupler 204. In the examples of FIG. 11, FIG. 12 and FIG. 13 the optical circuit 1102, 1208, 1302 comprises an optical coupler 1104 with N input ports and N output ports and plurality of other optical components.

[0242] The results presented in FIG. 9, FIG. 10 and FIG. 17 demonstrate that the measurement systems disclosed herein exhibit significantly reduced optical losses when compared with comparable prior art approaches (in particular a benchmark based on a Benes switching network 902). The reduction in optical loss (when compared with the Benes switching network 902 benchmark) generally increases as the size of the measurement system (number of input modes) increases. The advantages associated with measurement systems disclosed herein may therefore be particularly pronounced for relatively large measurement systems arranged to perform Bell state measurements (and generate quantum communication links) for relatively large quantum computing systems 102.

[0243] A reduction m optical loss, as has been shown to be realised by several examples disclosed herein, may serve to improve an efficiency with which Bell state measurements may be performed using a measurement system. For example, the proportion of Bell state measurements which are attempted by a measurement system and which are successful may be increased by reducing optical losses in the measurement system, which increases the proportion of photons which reach a measurement unit 508. Measurements systems as disclosed herein may therefore serve to reduce (on average) a number of attempts needed to perform a successful Bell state measurement and establish a quantum communication link.

[0244] Various examples of measurement systems 502, 602, 702 have been disclosed herein which include a switching network comprising a plurality of optical switches. Whilst not shown in the drawings, a measurement system 502, 602, 702 comprising an optical switching network may further comprise a switching controller. A switching controller may be configured to control the optical switching network (for example by controlling switches which form the switching network) so as to direct a photon received at any of a plurality of inputs to an input port of any of the optical couplers included in the measurement system. The switching controller may in particular control the switching network so that photons originating from two quantum computing nodes 104 between which a quantum communication link is to be established, are directed to input ports of the same optical coupler 204 such that a Bell state measurement of the photons may be performed (in order to establish the quantum communication link). The switching controller may receive instructions (for example, from a controller responsible for establishing quantum communication links between the quantum computing nodes 104). For example, the switching controller may receive instructions to facilitate establishment of a quantum communication link between two quantum computing nodes 104. The switching controller may, in response to the instructions, control the switching network to direct photons originating from the two quantum computing nodes 104 to the same optical coupler 204 such that a Bell state measurement of the photons may be performed (in order to establish the quantum communication link).

[0245] Various examples of measurement systems 202, 502, 602, 702, 1112, 1210, 1502, 1602 have been disclosed herein in which optical components such as one or more optical couplers, one of more optical switches and optical waveguides are included. All or part of any measurement system (including, for example, optical couplers, switches and / or optical waveguides) contemplated herein may be fabricated according to any suitable process. For example, all or part of any measurement system contemplated herein may be fabricated using silicon photonics components and / or fabrication processes.

[0246] Various examples, of measurement systems 202, 502, 602, 702, 1112, 1210, 1502, 1602 have been disclosed herein in which include a 4 x 4 optical coupler. In other examples, other N x N optical couplers may be used where N is greater than two but is not necessarily limited to four.

[0247] In the interest of conciseness not all possible alternatives which fall within the scope of the present disclosure have been explicitly discussed herein. As the skilled person will appreciate, in the present disclosure any aspect discussed from the perspective of an element being operable to do an action also discloses the same feature from the perspective of a method including a method step corresponding to the action. Similarly, any discussion presented from the perspective of a method step also discloses the same features from the perspective of any one or more suitable elements being operable or configured to carry out some or all of the method step. It is also considered within the present disclosure that for any method step(s), there can be a computer program configured to carry out, when executed, the method step(s).

[0248] Examples of the present disclosure can be realised in the form of hardware, software or a combination of hardware and software. Any such software may be stored in the form of volatile or non-volatile storage such as, for example, a storage device like a ROM, whether erasable or rewritable or not, or in the form of memory such as, for example, RAM, memory chips, device or integrated circuits or on an optically or magnetically readable medium such as, for example, a CD, DVD, magnetic disk or magnetic tape. It will be appreciated that the storage devices and storage media are embodiments of machine-readable storage that are suitable for storing a program or programs that, when executed, implement examples of the present disclosure. Accordingly, examples provide a program comprising code for implementing a system or method as claimed in any preceding claim and a machine readable storage storing such a program. Still further, examples of the present disclosure may be conveyed electronically via any medium such as a communication signal carried over a wired or wireless connection and examples suitably encompass the same.

[0249] Features, integers, characteristics, or groups described in conjunction with a particular aspect, embodiment or example of the invention or disclosure are to be understood to be applicable to any other aspect, embodiment or example described herein unless incompatible therewith. All of the features disclosed in this specification (including any accompanying claims, abstract and drawings), and / or all of the steps of any method or process so disclosed, may be combined in any combination, except combinations where at least some of such features and / or steps are mutually exclusive. The invention is not restricted to the details of any foregoing examples.

[0250] [1] “General self-imaging properties in N x N multimode interference couplers including phase relations”, Bachmann, M, Besse, P.A., Melchior, H., Applied Optics, 3905-3911, Vol. 33, No. 18, 20 June 1994.

[0251] [2] ‘‘Multimode quantum interference of photons in multiport integrated devices”, Peruzzo, A., Laing, A., Politi, A. et al. Nat Communications 2:224 (2011).

[0252] [3] “Optical Switch Based on Multimode Interference Coupler”, Wang, F., Yang, J., Chen, L., Jiang, X. and Wang, M., IEEE Photonics Technology Letters, vol. 18, no. 2, pp. 421-423, Jan. 15, 2006.

[0253] [4] ‘‘Analysis of Generalized Mach-Zehnder Interferometers for Variable-Ratio Power Splitting and Optimized Switching”, Nagali, N., Paiam, M., MacDonald, R., Worhoff, K. And Driessen A., Journal of Lightwave Technology, vol. 17, no. 12, pp. 2542-2550, Dec. 1999.

[0254] [5] “Optimal design of a 4 x 4 MM1 thermal optical switch with trapezoidal air trenches”, Shang Y„ Zhou J., Jiang H., He X., Ye X., Li C. . Appl Opt. 2023 Feb 20;62(6): 1521-1527.

[0255] [6] ‘‘Dilated Networks for Photonic Switching”, Padmanabhan K, Netravali, A., IEEE Transactions on Communications, Vol. 35, no. 12, pp. 1357-1365, December 1987.

Claims

1. A measurement system for a quantum computing system, the measurement system comprising:a first optical circuit comprising N input ports and N output ports, wherein N is greater than two, and wherein the N input ports are arranged to receive a first photon at a first of the N input ports and a second photon at a second of the N input ports, the first photon having quantum entanglement with a first qubit and the second photon having quantum entanglement with a second qubit and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports; anda first Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the N output ports, wherein the Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit.

2. The measurement system of claim 1, wherein the first optical coupler comprises a multimode interference coupler.

3. The measurement system of claim 1 or 2, wherein N is equal to four such that the first optical circuit comprises four input ports and four output ports.

4. The measurement system of any one of claims 1 to 3, wherein the Bell state measurement apparatus further comprises a measurement analyser arranged to receive data from the single photon detectors indicative of measurements of photons made by the single photon detectors, and wherein the measurement analyser is configured to determine a Bell state of the first photon and the second photon based on the data received from the single photon detectors.

5. The measurement system of any one of claims 1 to 4, wherein the plurality of single photon detectors are configured to measure photons in a plurality of time bins and wherein the Bell state measurement apparatus is configured to project the measurements made by the plurality of single photon detectors onto the Bell state based on the time bin and the single photon detector at which each of the first photon and the second photon are detected.

6. The measurement system of claim 5, wherein the Bell state measurement apparatus is configured to project the measurements made by the plurality of single photon detectors onto the Bell state further based on the input port at which each of the first photon and the second photon are received.

7. The measurement system of any one of claims 1 to 6, wherein the plurality of single photon detectors comprises N single photon detectors arranged to receive and measure photons output from each of the N output ports.

8. The measurement system of any one of claims 1 to 7, wherein N is greater than three , wherein the N input ports are arranged to receive a third photon at a third of the N input ports and a fourth photon at a fourth of the N input ports the third photon having quantum entanglement with a third qubit and the fourth photon having quantum entanglement with a fourth qubit, andwherein the optical circuit comprises the first optical coupler and a plurality of further optical couplers arranged to provide a plurality of optical paths from each of the N input ports of the optical circuit to at least one of the N output ports of the optical circuit, wherein each of the optical paths passes through the first optical coupler; andwherein the first Bell state measurement apparatus comprises N single photon detectors arranged to receive and measure photons output from the N output ports of the optical circuit, wherein the Bell state measurement apparatus is configured to project measurements made by the N single photon detectors onto Bell states so as to perform the first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit and a second Bell state measurement of the third photon and the fourth photon for generating quantum entanglement between the third qubit and the fourth qubit.

9. The measurement system of claim 8, wherein the plurality of further optical couplers comprises a plurality of 2 x 2 optical couplers each comprising two input ports and two output ports.

10. The measurement system of claim 8, wherein the optical circuit comprises at least one phase shifter arranged in at least one of the optical paths from the N input ports of the optical circuit to at least one of the N output ports of the optical circuit.

11. The measurement system of any one of claims 8 to 10, wherein the optical circuit is arranged such that when each of the first, second, third and fourth photons are in a superposition of an early time bin and a late time bin, measurements made by the single photon detectors can beprojected onto Bell states based on the time bins in which photons are detected and the single photon detectors at which the photons are detected.

12. The measurement system of any one of claims 8 to 11, wherein the first Bell state measurement apparatus and the second Bell state measurement are independent of each other and can be performed in parallel.

13. The measurement system of any one of claims 1 to 12, further comprising:a second optical circuit comprising M input ports and M output ports, wherein M is greater than two, and wherein the M input ports are arranged to receive a fifth photon at a first of the M input ports and a sixth photon at a second of the M input ports, the fifth photon having quantum entanglement with a fifth qubit and the sixth photon having quantum entanglement with a sixth qubit, wherein the second optical circuit comprises a second optical coupler comprising M input ports and M output ports;a second Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the M output ports, wherein the second Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a third Bell state measurement of the fifth photon and the sixth photon, for generating quantum entanglement between the fifth qubit and the sixth qubit; andan optical switching network comprising a plurality of inputs each arranged to receive a photon, wherein the optical switching network is operable to selectively direct a photon received at any of the plurality of inputs to an input port of the first optical circuit or an input port of the second optical circuit.

14. The measurement system of any one of claims 1 to 13, wherein the optical switching network comprises a switching controller configured to control the optical switching network so as to direct a photon received at any of the plurality of inputs to an input port of the first optical circuit or an input port of the second optical circuit.

15. The measurement system of claim 14, wherein the switching controller is configured to control the optical switching network to:direct the first photon received at a first of the plurality of inputs to the first of the N input ports of the first optical circuit;direct the second photon received at a second of the plurality of inputs to the second of the N input ports of the first optical circuit;direct the fifth photon received at a third of the plurality of inputs to the first of the M input ports of the second optical circuit; anddirect the sixth photon received at a fourth of the plurality of inputs to the second of the M input ports of the second optical circuit.

16. The measurement system of any one of claims 13 to 15, wherein the optical switching network comprises a plurality of optical switches, and wherein at least some of the plurality of optical switches have X inputs and X outputs, wherein X is greater than two.

17. The measurement system of claim 16, wherein the at least some of the plurality of optical switches comprise a multimode interference switch.

18. The measurement system of claim 17, wherein the multimode interference switch comprises a plurality of multimode interference couplers and at least one phase shifter situated in an optical path between the plurality of multimode interference couplers.

19. The measurement system of any one of claims 16 to 18, wherein the at least some of the plurality of optical switches comprise a plurality of optical switches having two inputs and two outputs, wherein the plurality of optical switches are arranged to function as an optical switch having X inputs and X outputs.

20. The measurement system of claim 19, wherein the plurality of optical switches having two inputs and two outputs are arranged to provide fully connected and non-blocking switching between each of the X inputs and each of the X outputs.

21. The measurement system of any one of claims 16 to 20, wherein X is equal to four such that the at least some of the optical switches have four inputs and four outputs.

22. A measurement system for a quantum computing system comprising:a plurality of measurement units comprising N input ports; andan optical switching network comprising a plurality of inputs each arranged to receive a photon, wherein the optical switching network is operable to selectively direct a photon received at any of the plurality of inputs to an input port of any of the plurality of measurement units, wherein each of the plurality of measurement units comprises:an optical circuit comprising the N input ports and N output ports, wherein N is greater than two and wherein the N input ports are arranged to receive a first photon at a first of the N input ports and a second photon at a second of the N input ports and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports; anda Bell state measurement apparatus comprising a plurality of single photon detectors arranged to receive and measure photons output from the N output ports, wherein the Bell state measurement apparatus is configured to project measurements made by the plurality of single photon detectors onto a Bell state so as to perform a Bell state measurement of the first photon and the second photon.

23. A quantum computing system comprising:a plurality of quantum computing nodes, each quantum computing node being configured to maintain at least one qubit situated at the quantum computing node and wherein each quantum computing node is operable to generate a photon having quantum entanglement with the at least one qubit; anda measurement system according to any one of claims 1 to 22, the measurement system being arranged to receive at least a first photon having quantum entanglement with a first qubit at a first of the plurality of quantum computing nodes and a second photon having quantum entanglement with a second qubit at a second of the plurality of quantum computing nodes, wherein the measurement system is operable to perform a first Bell state measurement of the first photon and the second photon for generating quantum entanglement between the first qubit and the second qubit.

24. A method of operating a measurement system for a quantum computing system, the method comprising:receiving a first photon at a first input port of an optical circuit, the optical circuit having N input ports and N output ports, wherein N is greater than two, and wherein the first optical circuit comprises a first optical coupler comprising N input ports and N output ports;receiving a second photon at a second input port of the optical circuit;measuring, with at least one of a plurality of single photon detectors arranged at a plurality of the N output ports of the optical circuit, the first photon and the second photon;projecting the measurements of the first photon and the second photon onto a Bell state so as to perform a first Bell state measurement of the first photon and the second photon.

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